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p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ , τo ), discussed in our last set of notes, we can describe a basis B for τo as follows: ¯ ¯ B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. The sets in B are τo -open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. 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. As we have just discussed, (SΩ , τ0 ) is zero-dimensional, as are the discrete and indiscrete topologies on any set. It can also be shown that the Sorgenfrey line (R, τs ) is zero-dimensional. Recall that a basis for τs is ¯ B = {[a, b) ¯ a, b ∈ R and a < b}. It is easy to show that each set [a, b) is clopen relative to τs : each [a, b) itself is τs -open by definition of τs , and the complement of [a, b) is (−∞, a)∪[b, ∞), [¡ ¢ [a − n, a) ∪ [b, b + n) , and is therefore open. which can be written as n∈N Closures and Interiors of Sets As you may know from studying analysis, subsets are frequently neither open nor closed. However, for any subset A in a topological space, there is a certain closed set A and a certain open set Ao associated with A in a natural way: \ ¯ 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. p. 2 Prop N7.1 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 N7.2 Let τ µ ∈ Π(X) and τ ≤ µ. Then: (a) Clµ A ⊆ Clτ A; (b) Iτ A ⊆ Iµ A. To illustrate Prop 2, consider the extreme cases τ = τi and µ = τd . If φ 6= A ( X, then Iτi A = φ ⊆ Iτd A = A = Clτd A ⊆ X = Clτi 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 N7.3 Let (X, τ ) be a topological space, with A ⊆ X. Then ¯ A = {x ∈ X ¯ ∀ nbhds U of x, A ∩ U 6= φ}. ¯ Proof : Define B = {x ∈ X ¯ ∀ nbhds U of x, A ∩ U 6= φ}. First note that A ⊆ B, since x ∈ A ⇒ (x ∈ U ∩ A for all nbhds U of x) ⇒ (U ∩ A 6= φ for all nbhds U of x) ⇒ x ∈ B. Now, if we can show that B is closed, we’ll have A ⊆ B, since A is the smallest closed set containing A. So, let y ∈ X − B. Then by assumption, there is a nbhd Vy of y such that Vy ∩ A = φ. Indeed, it also follows that Vy ∩ B = φ, for if there was a z ∈ Vy ∩ B, then Vy would be a nbhd of z, and since z ∈ B, we’d have (by definition of B) Vy ∩ A 6= φ, a contradiction. Thus, for all y ∈ X − B, there is a nbhd Vy of y such that Vy ∩ B = φ, or equivalently, Vy ⊆ X − B. Thus X − B is open (since it is the union of these Vy ’s). It remains to show B ⊆ A. Suppose x ∈ B − A. Then there is a closed set D containing A such that x 6∈ D. Thus x ∈ X − D ⊆ X − A, so U = X − D is a nbhd of x having an empty intersection with A, contrary to our assumption that x ∈ B. Thus B ⊆ A. Combining this with the previous result that A ⊆ B, we have A = B. ¥ p. 3 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 N7.3 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 N7.3, one can see that A′ ⊆ A. Prop N7.4 For any set A in a topological space, A = A ∪ A′ . Proof : We have already observed that A ∪ A′ ⊆ A. To prove the opposite inclusion, it suffices to show that (x ∈ A and x 6∈ A) =⇒ x ∈ A′ . If x ∈ A, then by Prop N7.3, U ∩ A 6= φ for all nbhds U of x. But now x 6∈ A =⇒ U ∩ (A − {x}) 6= φ for all nbhds U of x, and this implies that x ∈ A′ , as desired. ¥ The set A′ of all limit points of A is 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”. p. 4 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 N7.5 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 N7.3. Corollary For any set A (in a topological space), BdA = A − Ao . Remark: Prop N7.4 shows that for any set A, Ao = X − (X − A). Likewise, for any set B (which we may identify with X − A in Prop N7.4), B = X − (X − B)o . Thus the interior operator can be defined in terms of the closure operator, and vice-versa. Prop N7.6 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. p. 5 In the interest of aligning these notes a bit more with our text, I’ll mention the product and subspace topologies now, though we’ll revisit these concepts again soon. The Product Topology on X × Y (§15 in Munkres) Let (X, τ ) and (Y, µ) be topological spaces. The product topology on X × Y is the topology having basis ¯ B = {U × V ¯ U ∈ τ, V ∈ µ}. Prop N7.7 (Munkres, p.86) Referring to the previous definition, if Bτ is a basis for τ ¯ and Cµ is a basis for µ, then {B × C ¯ B ∈ Bτ , C ∈ Cµ } is a basis for the product topology on X × Y . The Subspace Topology (§16 in Munkres) Let (X, τ ) be a topological space, and let A ⊆ X. Then the collection of sets ¯ τA = {U ∩ A ¯ U ∈ τ } is a topology on A, called the subspace topology. (A, τA ) is called a (topological) subspace of (X, τ ). A is said to “inherit” its topology τA from its parent space (X, τ ). Prop N7.8 (Munkres, p.89) If (X, τ ) is a topological space with τ having basis B, and ¯ if A ⊆ X, then {U ∩ A ¯ U ∈ B} is a basis for the subspace topology τA on A. HW 3 (due Monday 9/12, any time): §16 (p. 91): 3, 8