Download 1 Metric Spaces

1
Metric Spaces
Definition 1.1. A metric on a set X is a function d : X × X → R such that
for any x, y and z in X,
(i) d(x, y) ≥ 0,
(ii) d(x, y) = d(y, x),
(iii) d(x, y) = 0 if and only if x = y,
(iv) d(x, y) ≤ d(y, z) + d(z, x).
“triangle inequality”
A metric space is a nonempty set X together with a metric d on it, usually
denoted by (X, d).
Definition 1.2. Let (X, d) be a metric space, p ∈ X and r a positive real
number. The d-open ball with center p and radius r is the set
Bd (p; r) = {x ∈ X | d(x, p) < r}.
Definition 1.3. Let (X, d) be a metric space X. A set V ⊆ X is said to be a
neighborhood of x if there is an ε > 0 such that Bd (x; ε) ⊆ V .
Denote by N(x) the collection of all neighborhoods of x.
Definition 1.4. Let G be a subset of a metric space (X, d). G is said to be
d-open if it is a neighborhood of each of its points. In other words, G is d-open
if for any x in G, there is an ε > 0 such that Bd (x; ε) ⊆ G. When the metric
d is understood, we may simply say that G is an open set.
Theorem 1.5. Any d-open ball is d-open.
Theorem 1.6. For any metric space (X, d),
(i) ∅ and X are d-open.
(ii) any union of d-open sets is d-open.
(iii) any finite intersection of d-open sets is d-open.
Definition 1.7. Let (X, d) be a metric space. The family of d-open sets in X
is called the topology for X generated by d.
Definition 1.8. Two metrics d and ρ on X are said to be equivalent if they
generate the same topology, i.e. τd = τρ .
Definition 1.9. A subset F of X is said to be d-closed if its complement
X − F is d-open.
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Corollary 1.10. For any metric space (X, d),
(i) ∅ and X are d-closed.
(ii) any intersection of d-closed sets is d-closed.
(iii) any finite union of d-closed sets is d-closed.
Definition 1.11. Let A and B be nonempty subsets of a metric space (X, d).
The distance between A and B, denoted by d(A, B), is defined to be
d(A, B) = inf{ d(a, b) | a ∈ A and b ∈ B }.
If B = { x}, then the distance between A and B is called the distance between
x and A and is denoted by d(x, A).
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2
Topological Spaces
Definition 2.1. A topology τ on a set X is a collection of subsets of X such
that
(i) ∅ and X belong to τ .
(ii) any union of elements of τ belongs to τ .
(iii) any finite intersection of elements of τ belongs to τ .
A topological space is a nonempty set X together with a topology τ on it,
usually denoted by (X, τ ). The elements of τ are called the open sets (of X).
Examples.
1. On any nonempty set X, the collection of all subsets of X is a topology
on X, called the discrete topology on X. Also, the collection { ∅,X} is also a
topology on X, called the indiscrete topology on X.
2. Let (X, d) be a metric space. The set of all d-open sets is a topology on X,
called the metric topology on X induced by d, denoted by τd .
Definition 2.2. Let (X, τ ) be a topological space. If there exists a metric d
on X such that τ = τd , we say that (X, τ ) is metrizable.
Definition 2.3. Let A be a subset of a topological space X. A set W ⊆ X is
said to be a neighborhood of A if there is an open set G such that A ⊆ G ⊆ W .
If A = {x}, a neighborhood of A is usually called a neighborhood of x.
Denote by N(x) the collection of all neighborhoods of x.
Theorem 2.4. Let x be an element of a topological space X.
(i) If W ∈ N(x), then x ∈ W .
(ii) If V , W ∈ N(x), then V ∩ W ∈ N(x).
(iii) If W ∈ N(x) and W ⊆ V , then V ∈ N(x).
Theorem 2.5. Let G be a subset of a topological space X. G is open if and
only if it is a neighborhood of each of its points.
Definition 2.6. Let A be a subset of a topological space X. The closure
of A (in X), denoted by A, is the set of all points x in X such that every
neighborhood of x meets A.
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Theorem 2.7. Let A and B be arbitrary subsets of a topological space X.
(i) If A ⊆ B, then A ⊆ B.
(ii) A ⊆ A.
(iii) A is the smallest closed set containing A.
(iv) A is closed if and only if A = A.
(v) A = A.
(vi) A ∪ B = A ∪ B.
Definition 2.8. Let A be a subset of a topological space X. A point x in X is
said to be an accumulation point or a limit point of A if every neighborhood of
x meets A − {x}. The set of all accumulation points of A is called the derived
set of A, denoted by A0 .
Theorem 2.9. A = A ∪ A0 . In particular, A is closed if and only if it contains
all its accumulation points.
Definition 2.10. A subset A of a topological space X is said to be dense (in
X) if A = X.
Theorem 2.11. Let A be a subset of a topological space X. A is dense if and
only if every nonempty open subset of X meets A.
Definition 2.12. Let A be a subset of a topological space X. A point x ∈ X
is said to be an interior point of A if A is a neighborhood of x. The set of all
interior points of A is called the interior of A, denoted by Int A.
A point x ∈ X is said to be an exterior point of A if x is an interior point
of X − A. The exterior of A, denoted by Ext A, is the set of all exterior points
of A.
The frontier or boundary of A is the set Fr A = A ∩ X − A.
Theorem 2.13. Let A be a subset of a topological space X. Then Int A,
Ext A and Fr A form a partition of X.
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3
Continuous Functions
Definition 3.1. Let f be a function from a topological space X into a topological space Y . Let xo ∈ X. f is said to be continuous at xo if, whenever W
is a neighborhood of f (xo ) in Y , f −1 [W ] is a neighborhood of xo in X.
f is said to be continuous (on X) if f is continuous at every point in X.
f is said to be a homeomorphism if f is 1-1, onto and both f and f −1 are
continuous. In this case, X and Y are said to be homeomorphic.
Theorem 3.2. Let f : X → Y and xo ∈ X. Then f is continuous at xo if
and only if for every neighborhood W of f (xo ), there is a neighborhood V of
xo such that f [V ] ⊆ W .
Theorem 3.3. Let (X, d) and (Y, ρ) be metric spaces, f : X → Y and xo ∈ X.
Then f is continuous at xo if and only if for every ε > 0, there is a δ > 0 such
that for every x ∈ X, d(x, xo ) < δ implies ρ(f (x), f (xo )) < ε.
Theorem 3.4. Let f : X → Y and g : Y → Z. If f is continuous at xo and g
is continuous at f (xo ), then g ◦ f is continuous at xo .
Theorem 3.5. Let F denote either the set of real numbers R or the set of
complex numbers C. Let f and g be continuous functions from a topological
space X into F. Prove that the functions f + g, f − g, f · g, |f | are continuous
and if g(x) 6= 0, for all x ∈ X, then f /g is also continuous.
Theorem 3.6. Let f : X → Y . Then the following statements are equivalent:
(a) f is continuous;
(b) If B is open in Y , then f −1 [B] is open in X;
(c) If B is closed in Y , then f −1 [B] is closed in X.
Definition 3.7. A function f : X → Y is said to be open (closed) if for any
open (closed) subset A of X, f [A] is open (closed) in Y .
Theorem 3.8. Let f : X → Y be 1-1 and onto. Then the following statements
are equivalent :
(a) f is a homeomorphism;
(b) f is continuous and open;
(c) f is continuous and closed.
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4
Subspaces
Theorem 4.1. Let (X, τ ) be a topological space and S ⊆ X. Define
τS = { G ∩ S | G ∈ τ }.
Then τS is a topology on S. Moreover, τS is the smallest topology on S which
makes i : S → X continuous, where i(x) = x for all x ∈ S is the inclusion
map.
Definition 4.2. Let (X, τ ) be a topological space and S ⊆ X. Then the
topology τS is called the relative topology on S and (S, τS ) is called a subspace
of X.
Theorem 4.3. Let S be a subset of a metric space (X, d). The restriction of
d to S × S is a metric on S, and the metric topology induced by d |S×S is the
relative topology on S.
Theorem 4.4. Let S be a subspace of a topological space X and A ⊆ S.
Then A is closed in S if and only if there is a closed set F in X such that
A = F ∩ S.
Theorem 4.5. Let S be a subspace of a topological space X, and x ∈ S. A
subset W ⊆ S is a neighborhood of x in S if and only if there is a neighborhood
V of x in X such that W = V ∩ S.
Theorem 4.6. Let S be a subspace of a topological space X and A ⊆ S.
Then the closure of A in S is A ∩ S where A is the closure of A in X.
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5
Sequences
Definition 5.1. A sequence on a nonempty set X is a function from N into
X. The function { (n, f (n)) | n ∈ N } will be denoted by (xn ).
Definition 5.2. A sequence (xn ) in a topological space X is said to converge
to xo ∈ X if for any neighborhood V of xo there is an N ∈ N such that xn ∈ V
for any n ≥ N .
Theorem 5.3. Any sequence in a metric space converges to at most one limit.
Theorem 5.4. Let A be a subset of a topological space X and x ∈ X. If
there is a sequence (xn ) of points in A which converges to x, then x ∈ A.
The converse holds if X is a metric space (first countable space).
Theorem 5.5. Let f : X → Y . If f is continuous at x ∈ X, then for any
sequence (xn ) in X which converges to x, the sequence (f (xn )) converges to
f (x) in Y .
The converse holds if X is a metric space (first countable space).
Definition 5.6. Let (xn ) be a sequence in X. A subsequence (xnk ) of (xn ) is
a sequence k 7→ xnk where (nk ) is a strictly increasing sequence in N.
Theorem 5.7. In a topological space X, if a sequence (xn ) converges to x,
then every subsequence of (xn ) also converges to x.
Definition 5.8. Let (xn ) be a sequence in a topological space X. A point
x ∈ X is called a cluster point or a limit point of a sequence (xn ) if every
neighborhood of x contains xn for infinitely many n’s.
Theorem 5.9. Let (xn ) be a sequence in a topological space. If (xn ) has a
convergent subsequence, then (xn ) has a cluster point.
The converse holds if X is a metric space (first countable space).
Definition 5.10. Let (X, d) be a metric space. A sequence (xn ) in X is called
a Cauchy sequence if for any ε > 0, there is an N ∈ N such that d(xm , xn ) < ε
for any m ≥ N , n ≥ N .
Theorem 5.11. Any convergent sequence in a metric space is a Cauchy sequence.
Theorem 5.12. If a Cauchy sequence in a metric space has a convergent
subsequence, then that sequence converges.
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6
Complete Metric Spaces
Definition 6.1. A metric space (X, d) is said to be complete if every Cauchy
sequence in X converges (to a point in X).
Theorem 6.2. A closed subset of a complete metric space is a complete
subspace.
Theorem 6.3. A complete subspace of a metric space is a closed subset.
Definition 6.4. Let A be a nonempty subset of a metric space (X, d). The
diameter of A is defined to be
diam(A) = sup{d(x, y) | x, y ∈ A}.
We say that A is bounded if diam(A) is finite.
Theorem 6.5. Let (X, d) be a complete metric space. If (Fn ) is a sequence
of nonempty closed subsets of X such that Fn+1 ⊆ Fn for all n ∈ N and
T
(diam(Fn )) converges to 0. Then ∞
n=1 Fn is a singleton.
Definition 6.6. Let f be a function from a metric space (X, d) into a metric
space (Y, ρ). We say that f is uniformly continuous if given any ε > 0, there
exists a δ > 0 such that for any x, y ∈ X, d(x, y) < δ implies ρ(f (x), f (y)) < ε.
Theorem 6.7. A uniformly continuous function maps Cauchy sequences into
Cauchy sequences.
Definition 6.8. Let f be a function from a metric space (X, d) into a metric
space (Y, ρ). We say that f is an isometry if d(a, b) = ρ(f (a), f (b)) for any a,
b ∈ X.
Theorem 6.9. An isometry is uniformly continuous and is a homeomorphism
from X onto f [X].
Theorem 6.10. Let A be a dense subset of a metric space (X, d). Let f be a
uniformly continuous function (isometry) from A into a complete metric space
(Y, ρ). Then there is a unique uniformly continuous function (isometry) g from
X into Y which extends f .
Definition 6.11. A completion of a metric space (X, d) is a pair consisting of
a complete metric space (X ∗ , d∗ ) and an isometry ϕ of X into X ∗ such that
ϕ[X] is dense in X ∗ .
Theorem 6.12. Every metric space has a completion.
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Theorem 6.13. A completion of a metric space is unique up to isometry. More
precisely, if {ϕ1 , (X1∗ , d∗1 )} and {ϕ2 , (X2∗ , d∗2 )} are two completions of (X, d),
then there is a unique isometry f from X1∗ onto X2∗ such that f ◦ ϕ1 = ϕ2 .
Definition 6.14. A function f : (X, d) → (X, d) is said to be a contraction
map if there is a real number k < 1 such that d(f (x), f (y)) ≤ k d(x, y) for all
x, y ∈ X.
Theorem 6.15. Let f be a contraction map of a complete metric space (X, d)
into itself. Then f has a unique fixed point.
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7
Compactness I
Definition 7.1. A cover or a covering of a topological space X is a family C
of subsets of X whose union is X. A subcover of a cover C is a subfamily of C
which is a cover of X. An open cover of X is a cover consisting of open sets.
Definition 7.2. A topological space X is said to be compact if every open
cover of X has a finite subcover. A subset S of X is said to be compact if S
is compact with respect to the subspace topology.
Theorem 7.3. A subset S of a topological space X is compact if and only if
every open cover of S by open sets in X has a finite subcover.
Theorem 7.4. A closed subset of a compact space is compact.
Definition 7.5. A topological space X is said to be Hausdorff if any two
distinct points in X have disjoint neighborhoods.
Theorem 7.6. If A is a compact subset of a Hausdorff space X and x ∈
/ A,
then x and A have disjoint neighborhoods.
Theorem 7.7. Any compact subset of a Hausdorff space is closed.
Theorem 7.8. A continuous image of a compact space is compact.
Corollary 7.9. Let f : X → Y is a bijective continuous function. If X is
compact and Y is Hausdorff, then f is a homeomorphism.
Theorem 7.10. A continuous function of a compact metric space into a metric
space is uniformly continuous.
Definition 7.11. A metric space (X, d) is said to be totally bounded or precompact if for any ε > 0, there is a finite cover of X by sets of diameter less
than ε.
Theorem 7.12. A subspace of a totally bounded metric space is totally
bounded.
Theorem 7.13. Every totally bounded subset of a metric space is bounded.
A bounded subset of Rn is totally bounded.
Theorem 7.14. A metric space is totally bounded if and only if every sequence
in it has a Cauchy subsequence.
Definition 7.15. A space X is said to be sequentially compact if every sequence in X has a convergent subsequence.
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Theorem 7.16. A metric space X is sequentially compact if and only if it is
complete and totally bounded.
Definition 7.17. Let C be a cover of a metric space X. A Lebesgue number
for C is a positive number λ such that any subset of X of diameter less than
or equal to λ is contained in some member of C.
Theorem 7.18. Every open cover of a sequentially compact metric space has
a Lebesgue number.
Definition 7.19. A space X is said to satisfy the Bolzano-Weierstrass property if every infinite subset has an accumulation point in X.
Theorem 7.20. In a metric space X, the following statements are equivalent:
(a) X is compact;
(b) X has the Bolzano-Weierstrass property;
(c) X is sequentially compact;
(d) X is complete and totally bounded.
Theorem 7.21 (Heine-Borel). A subset of Rn is compact if and only if it
is closed and bounded.
Corollary 7.22 (Extreme Value Theorem). A real-valued continuous
function on a compact space has a maximum and a minimum.
Theorem 7.23 (Bolzano-Weierstrass). Every bounded infinite subset of
Rn has at least one accumulation point (in Rn ).
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8
Bases and Subbases
Definition 8.1. Let (X, τ ) be a topological space. A subset B of τ is called a
base for τ if every element of τ is a union of elements of B.
Theorem 8.2. Let (X, τ ) be a topological space and B ⊆ τ . B is a base for
τ if and only if for each G ∈ τ and each x ∈ G, there is a B ∈ B such that
x ∈ B ⊆ G.
Corollary 8.3. Let B be a base for τ . A subset G of X is open if and only if
for each x ∈ G, there is a B ∈ B such that x ∈ B ⊆ G.
Corollary 8.4. If τ and τ 0 are topologies for a set X which have a common
base B, then τ = τ 0 .
Theorem 8.5. Let X be a nonempty set. A family B of subsets of X is
a base for some topology τ on X if and only if X = ∪B and for every two
members U and V of B and each point x in U ∩ V , there is a W ∈ B such
that x ∈ W ⊆ U ∩ V .
Definition 8.6. Let C be a collection of subsets of X. The topology generated
by C is the smallest topology on X in which every element of C is open. C is
called a subbase for that topology.
Theorem 8.7. Every base for a topology is also a subbase.
Theorem 8.8. Let (X, τ ) be a topological space and C ⊆ P(X). C is a
subbase for τ if and only if the set of finite intersections of elements of C is a
base for τ .
Definition 8.9. Let x be a point in a space X. A neighborhood base or a local
base at x is a set Bx of neighborhoods of x such that for each neighborhood V
of x, there is an element B in Bx such that B ⊆ V .
Theorem 8.10. Let (X, τ ) be a topological space and B ⊆ τ . Then B is a
base for τ if and only if for each x ∈ X, the set Bx = { B ∈ B | x ∈ B } is a
neighborhood base at x.
Definition 8.11. Let (X, ≤) be a linearly ordered set. The order topology
on X is the topology generated by all sets of the form { x ∈ X | x < a } or
{ x ∈ X | x > a } for some a ∈ X.
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9
Product Topology
Definition 9.1. The Cartesian product of the family {Xα | α ∈ Λ} is defined
to be
n
o
Y
[
Xα = x : Λ →
Xα x(α) ∈ Xα for each α ∈ Λ .
α∈Λ
α∈Λ
We usually denote x(α) by xα and call it the α-th coordinate of x.
Q
For each β ∈ Λ, the function Pβ : α∈Λ Xα → Xβ defined by Pβ (x) = xβ
Q
is called the projection of α∈Λ Xα on Xβ or the β-th projection map.
Definition 9.2. Let Xα be a topological space for each α ∈ Λ. The product
Q
Q
topology on α∈Λ Xα is the topology on α∈Λ Xα generated by
{Pα−1 [Oα ] | α ∈ Λ and Oα is open in Xα }.
Q
Theorem 9.3. The product topology on
Xα is the smallest topology in
Q
which every projection Pβ : α∈Λ Xα → Xβ is continuous.
Q
Theorem 9.4. A base of the product topology on Xα is the collection of
Q
Q
all subsets of Xα of the form α∈Λ Uα where Uα is open in Xα for each α
and Uα = Xα for all but finitely many α’s.
Theorem 9.5. A function f from a topological space X into a product space
Q
α∈Λ Xα is continuous if and only if Pα ◦ f is continuous for each α ∈ Λ.
Q
Theorem 9.6. Let α∈Λ Xα be a product space. For each α ∈ Λ, the projection map Pα is open.
Q
Theorem 9.7. The set { α∈Λ Gα | Gα is open in Xα for each α ∈ Λ } is a
Q
base for a topology on
Xα . This topology is called the box topology on
Q
Xα .
Theorem 9.8. The box topology is in general larger than the product topology. For a finite product space, both topologies are the same.
Theorem 9.9. Let (X1 , d1 ), (X2 , d2 ), . . . , (Xn , dn ) be metric spaces. Define
dp ((x1 , . . . , xn ), (y1 , . . . , yn )) = d1 (x1 , y1 ) + · · · + dn (xn , yn ),
Q
Q
where (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ ni=1 Xi . Then dp is a metric on ni=1 Xi and
Q
the metric topology induced by dp is the product topology for ni=1 Xi .
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10
Countability Axioms
Definition 10.1. The space (X, τ ) is said to be first countable or to satisfy the
first axiom of countability if for each x ∈ X, there is a countable neighborhood
base at x.
Theorem 10.2. Let A be a subset of a first countable space space X and
x ∈ X. Then x ∈ A if and only if there is a sequence (xn ) of points in A
which converges to x.
Theorem 10.3. Let f : X → Y be a function, where X is first countable.
Then f is continuous at x ∈ X if and only if for any sequence (xn ) in X which
converges to x, the sequence (f (xn )) converges to f (x) in Y .
Definition 10.4. The space (X, τ ) is said to be second countable or to satisfy
the second axiom of countability if there is a countable base for τ .
Theorem 10.5. If (X, τ ) is second countable, then it is first countable.
Definition 10.6. A space X is said to be separable if X contains a countable
dense subset.
Theorem 10.7. If (X, τ ) is second countable, then it is separable.
Definition 10.8. A space X is said to be Lindelöf if every open cover of X
has a countable subcover.
Theorem 10.9. If (X, τ ) is second countable, then it is Lindelöf.
Theorem 10.10. Let X be a metric space. Then the followings are equivalent:
(a) X is second countable.
(b) X is separable.
(c) X is Lindelöf.
Theorem 10.11. Any subspace of a first (second) countable space is also first
(second) countable.
Remark. A subspace of a separable space may not be separable. A subspace
of a Lindelöf space may not be Lindelöf.
Theorem 10.12. An open subspace of a separable space is separable. A
closed subspace of a Lindelöf space is Lindelöf.
Theorem 10.13. A subspace of a separable metric space is separable.
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Theorem 10.14. A continuous image of a separable space is separable. A
continuous image of a Lindelöf space is Lindelöf.
Remark. A continuous image of a first countable space may not be first
countable. A continuous image of a second countable space may not be second
countable.
Theorem 10.15. Let f : X → Y be an open continuous function, and X is
first (second) countable. Then f [X] is also first (second) countable.
Corollary 10.16. Separability, first countability and second countability are
preserved under a homeomorphism.
Theorem 10.17. A nonempty countable product space is first (second) countable if and only if each factor is first (second) countable.
Theorem 10.18. A nonempty countable product space is separable if and
only if each factor is separable.
Remark. The product of two Lindelöf spaces need not be Lindelöf.
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11
Separation Axioms
Definition 11.1. A topological space X is said to be a T0 –space if for any
two distinct points in X, there is a neighborhood of one not containing the
other.
Definition 11.2. A topological space X is said to be a T1 –space if for any two
distinct points in X, each point has a neighborhood which does not contain
the other.
Theorem 11.3. A space X is T1 if and only if each singleton is closed. Hence,
a space X is T1 if and only if each finite subset is closed.
Theorem 11.4. If x is an accumulation point of a subset A of a T1 –space,
then every neighborhood of x contains infinitely many points of A.
Definition 11.5. A space X is said to be a T2 –space or a Hausdorff space if
any two distinct points have disjoint neighborhoods.
Theorem 11.6. Any sequence in a Hausdorff space has at most one limit.
Theorem 11.7. A space X is Hausdorff if and only if the diagonal ∆ =
{(x, x) | x ∈ X} is closed in X × X.
Theorem 11.8. Let f and g be functions from a space X into a space Y . If f
and g are continuous and Y is Hausdorff, then the set {x ∈ X | f (x) = g(x)}
is closed in X.
Corollary 11.9. Let f and g be functions from a space X into a Hausdorff
space Y . If f and g are continuous and f = g on a dense subset of X, then
f = g on X.
Corollary 11.10. If f is a continuous function from a dense subset of a space
X into a Hausdorff space Y , then there exists at most one continuous extension
of f .
Definition 11.11. A T1 –space is said to be a T3 –space or a regular space if any
point and any closed set not containing that point have disjoint neighborhoods.
Theorem 11.12. For any space X, the following statements are equivalent:
(a) For any closed set A and any point x ∈
/ A, x and A have disjoint neighborhoods.
(b) For each x ∈ X and each neighborhood V of x, there exists an open set
G such that x ∈ G ⊆ G ⊆ V .
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(c) For each x ∈ X and a closed set A not containing x, there is a neighborhood V of x such that V ∩ A = ∅.
Definition 11.13. A T1 –space is said to be a T3 1 –space or a completely regular
2
space or a Tychonoff space if for any closed set A and any x ∈
/ A, there exists
a continuous function f : X → [0, 1] such that f (x) = 0 and f [A] = 1.
Definition 11.14. A T1 –space is said to be a T4 –space or a normal space if
any two disjoint closed sets have disjoint neighborhoods.
Theorem 11.15. A space is normal if and only if for each closed set A and
each neighborhood V of A, there is an open set G such that A ⊆ G ⊆ G ⊆ V.
Theorem 11.16. A compact Hausdorff space is normal.
Theorem 11.17. A regular Lindelöf space is normal.
Theorem 11.18 (Urysohn’s Lemma). For any space X, the following are
equivalent:
(a) Each pair of closed sets have disjoint neighborhoods.
(b) If A and B are disjoint closed sets, then there is a continuous function
f : X → [0, 1] such that f [A] = {0} and f [B] = {1}.
(c) If A and B are disjoint closed sets, then there is a continuous function
f : X → [a, b] such that f [A] = {a} and f [B] = {b}.
Theorem 11.19 (Tietze Extension Theorem). For any space X, the following are equivalent:
(a) Each pair of closed sets have disjoint neighborhoods.
(b) For any closed subset A of X, every continuous function f : A → R has
a continuous extension g : X → R. Furthermore, if f [A] ⊆ [a, b], then g
can be chosen so that g[X] ⊆ [a, b].
Theorem 11.20. If X is a Ti –space and i ≥ j, then X is a Tj –space.
Any metric space is a Ti –space for all i.
Theorem 11.21. Subspaces of Ti –spaces are Ti –spaces for i = 0, 1, 2, 3, 3 21 .
Closed subspaces of a T4 –space are T4 -spaces.
Theorem 11.22. Let f : X → Y be a homeomorphism and X is a Ti –space,
then Y is a Ti –space for i = 0, 1, 2, 3, 3 12 , 4.
Theorem 11.23. A nonempty product space is a Ti –space if and only if each
factor is a Ti –space, i = 0, 1, 2, 3, 3 12 . (A product of two normal spaces may
not be normal.)
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12
Connectedness
Definition 12.1. A topological space is said to be disconnected if it is the
union of two nonempty disjoint open sets. A space is connected if it is not
disconnected.
A subspace of a topological space is said to be connected if it is a connected
space with respect to the subspace topology.
Definition 12.2. Two nonempty subsets A and B of a space X are called
separated sets if A ∩ B = ∅ and B ∩ A = ∅.
Theorem 12.3. The following statements are equivalent for a topological
space X:
1. X is disconnected.
2. X is the union of two disjoint, nonempty closed sets.
3. X is the union of two separated sets.
4. X has a proper subset A which is both open and closed.
5. X has a proper subset A such that Fr(A) = ∅.
6. There is a continuous function from X onto a discrete two-point space
{a, b}.
Corollary 12.4. The following statements are equivalent for a topological
space X:
1. X is connected.
2. X is not the union of two disjoint, nonempty closed sets.
3. X is not the union of two separated sets.
4. The only subsets of X which are both open and closed are X and ∅.
5. X has no proper subset A such that Fr(A) = ∅.
6. Every continuous function from X into a discrete space is constant.
Definition 12.5. A subset I of R is said to be an interval if for any real
numbers x, y, if x, y belong to I, then any real number between x and y also
belongs to I.
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Theorem 12.6. Let I be a subset of R. Then I is an interval if and only
if I is one of the following sets: ∅, [a, b], [a, b), (a, b], (a, b), [a, ∞), (a, ∞),
(−∞, b], (−∞, b), R.
Theorem 12.7. A subset of R is connected if and only if it is an interval.
Theorem 12.8. Let A and B be subsets of a topological space X. If A
is connected and A ⊆ B ⊆ A, then B is connected. In particular, if A is
connected, then A is connected.
Theorem 12.9. A continuous image of a connected space is connected.
Theorem 12.10 (Intermediate Value Theorem). Let f be a continuous
real-valued function defined on a connected space X. If a and b are in the
range of f , then for any c between a and b, there is an x ∈ X such that
f (x) = c.
Theorem 12.11. Let {Aα | α ∈ Λ} be a family of connected subsets of a
T
S
topological space X such that α∈Λ Aα 6= ∅. Then α∈Λ Aα is connected.
Corollary 12.12. Let An be connected subsets of a topological space X for
S
each n ∈ N. If An ∩ An+1 6= ∅ for each n ∈ N, then ∞
n=1 An is connected.
Corollary 12.13. A topological space X is connected if and only if every two
points of X lie in a connected subset of X.
Definition 12.14. A component of a space X is a maximal connected subset
of X.
Theorem 12.15. A component of a space is a closed set (but may not be an
open set).
Theorem 12.16. The components of a topological space form a partition of
it.
Theorem 12.17. A nonempty connected subspace of X which is both closed
and open in X is a component.
Definition 12.18. A space X is said to be totally disconnected if every component of X is a singleton.
Theorem 12.19. A nonempty product space is connected if and only if each
factor is connected.
Corollary 12.20. The components of a nonempty product spaces are the
products of the components of the factors.
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Corollary 12.21. A nonempty product space is totally disconnected if each
factor is totally disconnected.
Definition 12.22. Let X be a topological space and a, b ∈ X. A path from a
to b is a continuous function f : [0, 1] → X such that f (0) = a and f (1) = b.
A space X is said to be path–connected if every pair of points of X can be
joined by a path in X.
A subspace A of X is path–connected if it is a path–connected space with
respect to the subspace topology.
Theorem 12.23. A path–connected space is connected.
Theorem 12.24. A continuous image of a path–connected space is path–
connected.
Theorem 12.25. A nonempty product of path–connected space is path–
connected if and only if each factor is path–connected.
Definition 12.26. A space is said to be locally connected if every neighborhood of a point x ∈ X contains a connected neighborhood of x. Equivalently,
A space is locally connected if every point has a neighborhood base consisting
of connected sets.
Theorem 12.27. A space is locally connected if and only if the components
of open subsets are open.
Corollary 12.28. The components of a locally connected space is open.
Corollary 12.29. A nonempty open subset of R is the union of a countable
family of open disjoint intervals.
Theorem 12.30. A topological space (X, τ ) is locally connected if and only
if τ has a base consisting of connected subsets.
Theorem 12.31. An open subspace of a locally connected space is locally
connected.
Theorem 12.32. A continuous open (or closed) function maps a locally connected space onto a locally connected space.
Corollary 12.33. Local connectedness is preserved under a homeomorphism.
Theorem 12.34. A nonempty product space is locally connected if and only if
each factor is locally connected and all but finitely many factors are connected.
Corollary 12.35. A nonempty finite product space is locally connected if and
only if each factor is locally connected.
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13
Quotient Spaces
Definition 13.1. Let f be a continuous function of a space X onto a set Y .
The quotient topology on Y induced by f is the largest topology on Y such
that f is continuous. The set Y with this topology is called a quotient space
of X, and f is called a quotient map.
Theorem 13.2. The quotient topology on Y induced by f is given by
τf = {G ⊆ Y | f −1 [G] ∈ τX }.
Theorem 13.3. Let f be a continuous function from X onto Y . If f is either
open or closed, then f is a quotient map.
Corollary 13.4. Suppose f : X → Y is a continuous function from X onto
Y , X is compact and Y is Hausdorff. Then f is a quotient map.
Theorem 13.5. Suppose that Y has the quotient topology induced by a map
f : X → Y . Let g be a function from Y into a space Z. Then g is continuous
if and only if g ◦ f is continuous.
Definition 13.6. Let X be a topological space and r an equivalence relation
on X. The quotient (identification) space of X determined by r is the set of
equivalence classes X/r with the topology induced by the canonical projection
π : X → X/r, where π(x) = [x] = { a ∈ X | a r x}.
Theorem 13.7. Let f be a continuous function from X onto Y . Let r be
an equivalence relation on X such that f is constant on each equivalence
class. Then there exists a unique continuous function g : X/r → Y such that
f = g ◦ π.
Theorem 13.8. Let f be a continuous function from X onto Y . Define
an equivalence relation rf on X by a rf b if and only if f (a) = f (b). Let
g : X/rf → Y be the unique continuous function such that f = g ◦ π. Then g
is a homeomorphism if and only if f is a quotient map.
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14
Nets
Definition 14.1. A directed set is a pair {D, } where D is a set and is a
relation on D such that
(i) λ λ for every λ ∈ D.
(ii) For any α, β, γ ∈ D, if α β and β γ, then α γ.
(iii) For any α, β ∈ D, there is some γ ∈ D such that γ α and γ β.
Definition 14.2. Let X be a set. A net in X is a function of a directed set
{D, } into X. The net { (λ, xλ ) | λ ∈ D} will be denoted by (xλ )λ∈D .
Definition 14.3. A net (xλ )λ∈D in a topological space X is said to converge
to a point x if for any neighborhood V of x, there is an element λV ∈ D such
that for any λ ∈ D, λ λV implies xλ ∈ V . We call x a limit of the net
(xλ )λ∈D .
Theorem 14.4. Let A be a subset of a topological space X and x ∈ X. Then
x ∈ A if and only if there is a net in A which converges to x.
Corollary 14.5. Let A be a subset of a space X. A point x ∈ X is an
accumulation point of A if and only if there is a net in A−{x} which converges
to x.
Theorem 14.6. Let f : X → Y . Then f is continuous at x ∈ X if and only if
for any net (xλ )λ∈D in X, (xλ )λ∈D converges to x implies (f (xλ ))λ∈D converges
to f (x).
Q
Theorem 14.7. A net (xλ )λ∈D converges in a product space α∈Λ Xα to a
point b if and only if (Pα (xλ ))λ∈D converges to bα for each α ∈ Λ.
Theorem 14.8. A space X is Hausdorff if and only if every net in X converges
to at most one limit.
Definition 14.9. Let f : D → X be a net; let f (λ) = xλ . If M is a directed
set and g : M → D is a function such that
(i) µ1 µ2 implies g(µ1 ) g(µ2 ),
(ii) for each λ ∈ D, there is some µ ∈ M such that g(µ) λ,
then the composite function f ◦ g : M → X is called a subnet of (xλ )λ∈D . For
µ ∈ M , the net f ◦ g(µ) is denoted by (xλµ )µ∈M .
Theorem 14.10. If the net converges to a point x, then so does any subnet.
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Definition 14.11. We say that x ∈ X is a cluster poin of a net (xλ )λ∈D if for
each neighborhood U of x and for each λo ∈ D there is some λ λo such that
xλ ∈ U .
Theorem 14.12. A net has a cluster point if and only if it has a convergent
subnet.
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15
Compactness II
Theorem 15.1. Let X be a topological space. The following statements are
equivalent:
(a) X is compact.
(b) Every net in X has a cluster point.
(c) Every net in X has a convergent subnet.
Definition 15.2. A space X is said to be countably compact if every countable
open cover of X has a finite subcover.
Theorem 15.3. Let X be a topological space. The following statements are
equivalent:
(a) X is countably compact.
(b) Every countable family of closed subsets of X with the finite intersection
property has a nonempty intersection.
(c) Every sequence in X has a cluster point.
Theorem 15.4. A continuous image of a countably compact space is countably compact. A closed subspace of a countably compact space is countably
compact.
Theorem 15.5. If X is compact, then X is countably compact. The converse
holds if X is Lindelöf.
Theorem 15.6. If X is sequentially compact, then X is countably compact.
The converse holds if X is first countable.
Corollary 15.7. For a second countable space, compactness, countable compactness and sequential compactness are equivalent.
Corollary 15.8. For a metric space, compactness, countable compactness and
sequential compactness are equivalent.
Theorem 15.9 (Tychonoff ’s theorem). A nonempty product space is compact if and only if each factor is compact.
Definition 15.10. A space X is said to be locally compact if every point has
a compact neighborhood.
Theorem 15.11. Let X be a locally compact Hausdorff space. Then for any
x ∈ X, every neighborhood of x contains a compact neighborhood of x.
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Theorem 15.12. A locally compact Hausdorff space is completely regular.
Theorem 15.13. A closed subset of a locally compact space is locally compact. An open subset of a locally compact Hausdorff space is locally compact.
Theorem 15.14. If f is a continuous open map from X onto Y and X is
locally compact, then so is Y .
Theorem 15.15. A nonempty product space is locally compact if and only if
each factor is locally compact and all but finitely many factors are compact.
Corollary 15.16. A nonempty finite product is locally compact if and only
if each factor is locally compact.
b h} consisting
Definition 15.17. A compactification of a space X is a pair {X,
b and a homeomorphism h of X into X
b such that h[X] is
of a compact space X
b
dense in X.
b = X ∪ {ω}. Define
Theorem 15.18. Let X be a space and ω ∈
/ X. Let X
τb = τ ∪ { G ∪ { ω} | G ∈ τ and X − G is compact }
b with the following properties:
Then τb is a topology on X
b
(i) X is a subspace of X.
b τb) is a compact space.
(ii) (X,
b if and only if X is not compact.
(iii) X is dense in X
b is Hausdorff if and only if X is locally compact Hausdorff.
(iv) X
b i} is called a one-point
Definition 15.19. If X is not compact, then {X,
compactification or an Alexandroff compactification of X.
Theorem 15.20. A one-point compactification is unique up to homeomorphism.
Theorem 15.21 (Urysohn’s Lemma, LCH version). Let X be a locally
compact Hausdorff space. Let K be compact and G open subsets of X such
that K ⊆ G. Then there exists a continuous function f : X → [0, 1] such that
f [K] = {1} and f = 0 outside a compact subset of G.
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