Download p. 1 Math 525 Notes on section 17 Isolated points In general, a point

p. 1
Math 525 Notes on section 17
Isolated points
In general, a point x in a topological space (X, τ ) is called an isolated point iff the set {x}
is τ -open. A topological space is called discrete iff every point in the space is isolated.
Closed Sets
A subset A in a topological space (X, τ ) is defined to be closed (or τ -closed) iff X − A ∈ τ .
¯
That is, A is closed iff X − A is open. Let γτ = {A ⊆ X ¯ X − A ∈ τ } be the collection of
all τ -closed sets. Using the DeMorgan laws, it is relatively easy to see that γτ satisfies the
following three axioms, which are ”dual” to the open-set axioms:
(1) φ, X ∈ γτ ;
\
¯
Ai ∈ γτ ;
(2) If {Ai ¯ i ∈ I} ⊆ γτ , then
i∈I
(3) If A, B ∈ γτ , then A ∪ B ∈ γτ .
More generally, any collection of subsets of X satisfying these three ”closed set axioms” will
define the collection of closed sets for some topology on X; namely, the topology consisting
of the complements of the closed sets.
Prop. A8 Let (X, τo ) be any totally ordered set with its order topology. If a ≤ b in X, then
[a, b] is a τo -closed set.
Proof : Exercise.
Zero Dimensional Spaces
Sets which are simultaneously open and closed relative to the same topology are called
clopen sets. A topology with a basis of clopen sets is defined to be zero-dimensional.
The discrete and indiscrete topologies on any set are zero-dimensional. The Sorgenfrey line
(R, τs ) is also zero-dimensional.
Closures and Interiors of Sets
\ ¯
Clτ A = A = {B ¯ B is closed and A ⊆ B} (Closure of A)
[ ¯
Iτ A = Ao = {U ¯ U is open and U ⊆ A}. (Interior of A)
The notations Clτ A and Iτ A are used more commonly by topologists, who often want to
compare closures and interiors relative to different topologies. The shorter notations A and
Ao are favored by analysts, who consider different topologies on only rare occasions. The
following assertions are easy consequences of the preceding definitions.
Prop A9 Let A be a set in a toplogical space (X, τ ).
(a) A is the smallest closed overset of A;
(c) A is closed iff A = A;
(b) Ao is the largest open subset of A;
(d) A is open iff A = Ao .
Prop A10 Let τ , µ ∈ Π(X) and τ ≤ µ. Then:
(a) Clµ A ⊆ Clτ A;
(b) Iτ A ⊆ Iµ A.
To actually find A in a given situation, intersecting all the closed oversets of A is not a very
practical. The next proposition gives another approach.
Prop A11 Let (X, τ ) be a topological space, with A ⊆ X. Then
¯
A = {x ∈ X ¯ ∀ nbhds U of x, A ∩ U 6= φ}.
Proof :
Intuitively, one may think of A as consisting of points which are either in A or ”arbitrarily
close to” A. In topological spaces (unlike metric and normed spaces) there is no definition
of ”distance” to measure ”closeness”, but we can regard a point x as being ”arbitrarily close
to” a set A if every nbhd U of x intersects A.
Remark: The following slightly different version of Prop A11 is also valid:
¯
A = {x ∈ X ¯ for some basic nbhd system Bx of x, U ∩ A 6= φ for all U ∈ Bx }.
Limit Points of Sets
Let (X, τ ) be a topological space. Let A ⊆ X and y ∈ X. We define y to be a limit
point of A iff for every nbhd U of y, U ∩ (A − {y}) 6= φ. Using our previous interpretation
of ”arbitrarily close to”, we can say that y is a limit point of A iff y is ”arbitrarily close
to” points in A other than itself. Now, let A′ denote the set of all limit points of A. By
comparing the definition of limit point with Prop , one can see that A′ ⊆ A.
Thm 17.6 For any set A in a topological space, A = A ∪ A′ .
Proof : In Munkres.
The set A′ of all limit points of A is sometimes called the derived set of A. All the points
in the closure A are called closure points of A. So we can think of the closure points of A
as those points in X which are ”arbitrarily close to” A, and the limit points of A as those
points which are ”arbitrarily close to” A − {x}. If x ∈ X is such that {x} is open, then it
should be clear that x can not be a limit point of any set in X, which helps explain why
such points are called ”isolated points”.
The Boundary of a Set
The boundary of a set A is defined to be
BdA = A ∩ (X − A).
The boundary (as an intersection of two closed sets) is thus always a closed set, and a boundary point must be ”arbitrarily close to” A and X − A. Like closure points and limit points,
a boundary point of A may or may not belong to A.
Prop A12 Let (X, τ ) be a topological space with A ⊆ X. Then X − Ao = X − A.
Proof : (x 6∈ X − Ao ) iff (x ∈ Ao ) iff (∃ a nbhd U of x such that U ⊆ A) iff (x 6∈ X − A),
where the last ”iff” follows by Prop A11.
Corollary For any set A (in a topological space), BdA = A − Ao .
Remark: Prop A12 shows that for any set A, Ao = X − (X − A). Likewise, for any set B
(which we may identify with X − A in Prop A12), B = X − (X − B)o . Thus the interior
operator can be defined in terms of the closure operator, and vice-versa.
Prop A13 For any set A in a topological space, A = A ∪ (BdA).
Proof : (x ∈ A and x 6∈ A) =⇒ (x ∈ A − Ao ) =⇒ (x ∈ BdA). Thus A ⊆ A ∪ (BdA).
Conversely, let x ∈ A ∪ (BdA). Since (x ∈ A =⇒ x ∈ A) and (x ∈ BdA =⇒ x ∈ A), it
follows that A ∪ (BdA) ⊆ A.
¥
Note that A can be obtained by adding to A either its boundary points or its limit points,
though the notions of boundary points and limit points are generally different.
Convergence, and Hausdorff and T1 Spaces
In a topological space (X, τ ), we say a countable sequence of points {xn } converges (or τ converges) to a point x ∈ X iff for each τ -nbhd U of x, there exists a corresponding positive
integer N such that xn ∈ U for all n ≥ N . In familiar spaces such as R and R2 with the
usual topology, sequences can converge to at most one point. However, this is not true in a
general topological space. At this point, one might be asking, “For what types of topological
spaces is it true that sequences can converge to only one point?”. The answer lies in the
following discussion of the so-called Hausdorff and T1 properties.
A topological space (X, τ ) is called a Hausdorff space (or is said to be Hausdorff, or to
have the Hausdorff property) iff for every distinct pair of points x1 and x2 in X, there exists
a nbhd U1 of x1 and a nbhd U2 of x2 such that U1 ∩ U2 = φ. On page 99, Munkres proves
the following:
In a Hausdorff space X, every finite point set is closed.
The property that every finite point set is closed is not equivalent to being Hausdorff, however. For example (as Munkres points out), in the space (R, τcof ), every finite point set is
closed, but this space is not Hausdorff. This condition that all finite point sets be closed
is called the T1 axiom (or “condition”, or “property”). The relationship between these
properties (Hausdorff and T1 ) and convergence is addressed in the following theorems.
Munkres, Thm 17.9 Let (X, τ ) be a T1 topological space, and let A ⊆ X. Then the point x
is a limit point of A iff every τ -nbhd of x contains infinitely many points of A.
Munkres, Thm 17.10 If (X, τ ) is a Hausdorff space, then a sequence of points of X τ converges to at most one point of X.
There is generally much more focus on Hausdorff spaces than on T1 spaces, since every Hausdorff space is also T1 , and it is the property of convergence of sequences to at most one point
that is shared by most spaces that are encountered in application.