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P. 1 Math 525 Notes on Section 17 Isolated Points in General, a Point X

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P. 1 Math 525 Notes on Section 17 Isolated Points in General, a Point X 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 ; (2) If {Ai ¯ i ∈ I} ⊆ γτ , then \ Ai ∈ γτ ; ¯ 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) ¯ o Iτ A = A = [{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) ClµA ⊆ Clτ A; (b)Iτ A ⊆ IµA. 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 bound- ary 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 2 integer N such that xn ∈ U for all n ≥ N. In familiar spaces such as R and R 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, how- ever. 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)bea 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 Haus- dorff 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..
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