Download Contents 1. Topological Space 1 2. Subspace 2 3. Continuous

TOPOLOGY
Contents
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Topological Space
Subspace
Continuous Functions
Base
Separation Axiom
Compact Spaces and Locally Compact Spaces
Connected and Path connected
Product topology
Quotient Topology
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1. Topological Space
Let X be a nonempty set and P(X) be the set of all subsets of X. A family T ⊂ P(X)
is a topology on X if
(1) ∅, X ∈ T
(2) any union of members of T is a member of T
(3) any finite intersection of members of T is a member of T .
Elements of T are called open sets of X. A topological space is a pair (X, T ) where X is a
nonempty set and T is a topology on X. A topological space (X, T ) is simply denoted by
X when the topology T is specified. If U is an open set of X, we also say U is open in X.
A subset F is closed in X if X \ F is open in X.
Proposition 1.1. Let X be a topological space.
(1) ∅ and X are both closed sets.
(2) Any finite union of closed sets in X is closed.
(3) Any intersection of closed sets in X is closed.
Let X be a topological space. If U is an open set of X containing a point x ∈ X, U is
called an open neighborhood of x. Let A be a subset of X and x ∈ A. We say that x is an
interior point of A if there exists an open set U ⊂ A so that U is an open neighborhood of
x. The set of interior points is denoted by A◦ .
Proposition 1.2. A subset A of X is open if and only if A = A◦ . Moreover, A◦ is the
largest open set contained in A.
A point x ∈ X is an adherent point of a subset A of X if the intersection of A and any
open neighborhood of x is nonempty. The closure A of A is the set of all adherent points
of A.
Lemma 1.1. A is closed if and only if A = A. A is the smallest closed set containing A; it
is the intersection of all closed sets containing A.
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CALCULUS
A point x ∈ X is a boundary point of A ⊂ X if it is an adherent point of both of A and
X \ A. The topological boundary ∂A consists of all boundary points of A.
Theorem 1.1. A is the union of A◦ and ∂A.
2. Subspace
Let X be a topological space and Y be a nonempty subset of X. Let TY be the subset of
Y of the forms U ∩ Y for all U open in X.
Lemma 2.1. The family TY of subsets of Y forms a topology on Y.
The topology TY defined above is called the subspace topology of Y induced from X.
The pair (Y, TY ) or the symbol Y is called a topological subspace of X. Elements of TY are
also called relative open subsets of Y.
3. Continuous Functions
Let X and Y be topological spaces. A function f : X → Y is continuous if and only if
f −1 (V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous
at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such
that f (U ) ⊂ V.
Proposition 3.1. A function f : X → Y is continuous if and only if it is continuous at
every point of X.
Proposition 3.2. Let f : X → Y and g : Y → Z be continuous functions. Then g ◦ f :
X → Z is also continuous.
A bijection h : X → Y between spaces is a homeomorphism if both h, h−1 are continuous.
We say that X is homemorphic to Y or topologically equivalent to Y if there exists a
homeomorphism h from X onto Y.
Lemma 3.1. Topological equivalence is an equivalence relation.
4. Base
Let X be a topological space. A family B of open sets of X is a base for the topology of
X is every open set of X is a union of members of B.
Theorem 4.1. Let B be a family of open sets of X. The followings are equivalent.
(1) B is a base for the topology of X.
(2) For each x ∈ X and each open neighborhood U of x, there exists V ∈ B such that
V is an open neighborhood of X.
Not every family of subsets of X can be a base for a topology. The following theorem is
a characterization of a base for a topology on X.
Theorem 4.2. Let X be a nonempty set and B be a family of subsets of X. Then B is a
base for a topology on X if and only if the following two properties holds:
(1) each x lies in at least member of B,
(2) if U, V ∈ B, x ∈ U ∩ V, there exists W ∈ B such that W ⊂ U ∩ V.
Definition 4.1. A topological space X is second countable if there is a countable family
of open sets that forms a base for the topology for X.
CALCULUS
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S An open cover of a topological space X is a family of open sets U = {Uα : α ∈ Λ} so that
α Uα = X. An open subcover V of U is a subset of U such V is also an open cover of X.
Theorem 4.3. Let X be a second countable topological space. Then every open cover of
X has a countable subcover.
We say that a subset A of X is dense if A = X. A topological space X is separable if
there is a subset S that is countable and dense in X.
Theorem 4.4. If X is a second countable topological space, then X is separable.
5. Separation Axiom
Let X be a topological space.
(1) X is T1 if for any x, y ∈ X with x 6= y, there exists an open neighborhood U of y
such that x 6∈ U.
(2) X is T2 or Hausdorff if for any x, y ∈ X with x 6= y, there exist an open neighborhood
U of x and an open neighborhood V of x such that U ∩ V = ∅.
(3) X is regular if for each closed subset E of X and each x ∈ X \ E, there exist disjoint
open sets U, V such that x ∈ U and y ∈ V.
(4) X is T3 if X is a regular T1 -space.
(5) X is normal if for each pari E and F of disjoint closed sets of X, there exist disjoint
open sets U, V of X such that E ⊂ U and V ⊂ F.
(6) A T4 -space is a normal T1 -space.
Lemma 5.1.
T4 =⇒ T3 =⇒ T2 =⇒ T1 .
Theorem 5.1. Every metric space is a T4 -space.
Lemma 5.2. X is normal if and only if for each closed subset E of X and each open set
W containing E, there exists an open set U containing E such that U ⊂ W.
Theorem 5.2. Let E and F be disjoint closed subsets of a normal space X. There exists
a continuous function f : X → [0, 1] such that f = 0 on E and f = 1 on F.
Theorem 5.3. Let X be a normal space and Y be a closed subset of X. Suppose f is
a bounded continuous real-valued function on Y. There exists a bounded continuous realvalued function F on X such that F = f on Y.
6. Compact Spaces and Locally Compact Spaces
A space is compact if every open cover has finite subcover. A subset K of a space X is
a compact subset if it is a compact space with the subspace topology induced from X.
Proposition 6.1. Any finite union of compact subsets of a space is compact.
Proposition 6.2. A closed subspace of a compact space is compact.
Lemma 6.1. Let S be a compact subset of a space X. For each x ∈ X \ S, there exist
disjoint open neighborhood U of x and V of S.
Corollary 6.1. A compact subset of a Hausdorff space is closed.
Theorem 6.1. A compact Hausdorff space is normal.
Theorem 6.2. Let f : X → Y be a continuous function from a compact space X to a
space Y. Then f (X) is a compact subset of Y.
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CALCULUS
Theorem 6.3. Let f be a continuous function from a compact space X to a Hausdorff
space. If f is injective, then f is a homeomorphism of X and f (X).
A space X is locally compact if for each p ∈ X, there is an open neighborhood W of p
such that W is compact.
Example 6.1. Let Rn be equipped with the standard Euclidean topology. Then Rn is
locally compact Hausdorff space.
Theorem 6.4. Let X be a locally compact Hausdorff space and Y be a set consisting of
X and one other element. Then there exists a unique topology for Y such that Y becomes
compact Hausdorff space and the relative topology for X inherited from Y coincides with
the original topology for X.
Definition 6.1. The space Y constructed in the Theorem 6.4 is called the one point compactification of X.
7. Connected and Path connected
A space is disconnected if there exist open sets U, V such that
(1) X = U ∪ V
(2) U ∩ V = ∅
(3) U 6= ∅, and V 6= ∅.
Proposition 7.1. The only nonempty closed and open subset of a connected space is the
space itself.
A subset of a space is a connected subset if it is connected in the subspace topology.
Theorem 7.1. Let f : X → Y be a continuous function from a connected space into a
space. Then f (X) is connected.
Theorem 7.2. Let {ES
α } be a family of connected subsets of a space X such that Eα ∩Eβ 6=
∅ for each α, β. Then α Eα is connected.
Let X be a space and x ∈ X. The connected component C(x) of x is the maximal
connected subset of X containing x.
Proposition 7.2. Two connected components of X either coincide or are disjoint. The
connected components of X form a partition of X.
A path in X from x0 to x1 is a continuous function
γ : [0, 1] → X
such that γ(0) = x0 and γ(1) = x1 . We also say that x0 and x1 is connected by γ. The
space X is connected if any of its two points can be connected by a path.
Lemma 7.1. The relation “there is a path in X from x to y” is an equivalence relation on
X.
The equivalent classes corresponding to the above equivalence relation are called path
components of X.
Theorem 7.3. A path connected space is connected.
CALCULUS
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8. Product topology
Q Let {Xα : α ∈ Λ} be a collection of spaces and X be product of sets Xα , i.e. X =
α∈Λ Xα . Denote πα : X → Xα by (xα ) 7→ xα called the projection from X to Xα . The
product topology for X is the smallest
Q topology for each each projection πα is continuous.
In this section, X always stands for α Xα with the product topology.
Theorem 8.1. Let E be a space and f : E → X be a function. Then f is continuous if
and only if π ◦ f is continuous for all α.
Theorem 8.2. (Tychonoff’s Theorem )Any product of compact spaces is compact.
Now, let us consider finite product spaces. Suppose X1 , · · · , Xn are spaces.
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Theorem 8.3. The projections πj : nj=1 Xj → Xj are open mapping for all j.
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Theorem 8.4. If X1 , · · · , Xn are Hausdorff space, then ni=1 Xi is Hausdorff.
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Theorem 8.5. If X1 , · · · , Xn are path connected space, then ni=1 Xi is path connected.
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Theorem 8.6. If X1 , · · · , Xn are connected space, then ni=1 Xi is connected..
9. Quotient Topology
Let X be a topological space and R be an equivalence relation on X. The R-equivalence
class of x ∈ X is denoted by [x]. We denote by X/R the set of all R-equivalence classes of
elements of X. The function π : X → X/R sending an element x to its equivalence class [x]
is called the quotient map. Let T |X/R be the family of subsets U of X/R such that π −1 (U )
is open in X.
Lemma 9.1. TX/R is a topology on X/R. The set X together with TX/R is called the
quotient space of X. In this case, TX/R is called the quotient topology of X with respect to
R. It is the largest topology such that π : X → X/R is continuous.
Theorem 9.1. Let Y be a topological space, X, R, X/R, π be as above. A function f :
X/R → Y is continuous if and only if f ◦ π is continuous.
Theorem 9.2. Let f : X → Y be a continuous function. Let R be an equivalence relation
on X such that f is constant on each equivalent class. Then there exists a continuous
function g : X/R → Y such that f = g ◦ π.
Theorem 9.3. Let X, Y be compact Hausdorff space and f : X → Y be a continuous
surjection. Define a relation R as follows: We say x1 Rx2 if f (x1 ) = f (x2 ). Then R is an
equivalence relation. Equip X/R with the quotient topology. Then X/R is homeomorphic
to Y.