Some highlights on Countability and Separation Properties | 22M 132, Study notes of Topology

Download Some highlights on Countability and Separation Properties | 22M 132 and more Study notes Topology in PDF only on Docsity! 22M:132 Fall 07 J. Simon Some highlights on Countability and Separation Properties (Relates to text Sec. 30–36) Introduction. There are two basic themes to the next several sections: a. What properties of a topology allow us to conclude that the topology is given by a metric? b. What properties of a space allow us to conclude that the space actually is (homeomorphic to) a subspace of Rn (or at least a subspace of Rω)? Countability Properties. Here are several properties of spaces, all saying that the topology, or some key feature of it, can be described in terms of countably many pieces of information. The names are historical; they are not very descriptive or otherwise useful, but you should know them since they are used in the literature. (1) “First axiom of countability” The space (X, T ) is called first-countable if the topology has a countable local basis at each point x ∈ X. (2) ”Second axiom of countability” The space (X, T ) is called second-countable if the topology T has a countable basis. (3) ”Separable” The space (X, T ) is called separable if X contains a countable dense subset. Recall a subset A ⊆ X is called dense in X if the closure Ā is all of X, i.e. each open set contains at least one point of A. (4) Lindelöf property The space (X, T ) is called a Lindelöf space if each open cover of X has a countable sub-cover. The familiar space Rn, with the standard topology has all of the above properties (proof below). For more general spaces, we can ask many questions: • Do any of these properties imply others? • If a space X has one of the properties, do all subspaces of X have the property? (In that always happens, we would call the property hereditary.) • If we have a family of spaces with one of these properties, does the cartesian product have the property? • If f : X → Y is a continuous surjection, and X has one of the properties, must Y also have the property? • If (X, T ) has one of these properties, and T ′ is a coarser [resp. finer] topology, must (X, T ′) have the property? We will focus on just some highlights. c©J. Simon, all rights reserved page 1 Theorem (text 30.3). Countable basis =⇒ all the other countability properties. Proof. Suppose B is a countable basis for the topology on X. a. Countable local basis: Let x ∈ X and let U be any neighborhood of x. Since B is a basis for the topology, U is a union of elements of B. Thus there exists an element B ∈ B such that x ∈ B ⊆ U . So the set B is a countable local basis for each point x ∈ X. b. Separable: For each nonempty set B ∈ B, pick a point xB ∈ B. Since B is countable, the set {xB | B ∈ B} is countable. Since each open set is a union of elements of B, each nonempty open set U contains at least one of the sets B and so xB ∈ U . Thus {xB | B ∈ B} is dense in X. c. Lindelöf : Let {Uα}α∈J be an open cover of X. We want to prove there exists a countable subcover, by somehow using the existence of a countable basis B for the topology. For convenience (to make the exact argument a little simpler), assume that one of the sets Uα is actually the empty set. (Or adjoin one additional set U0 = ∅ to the covering.) The idea of the proof is to use the elements of B to “point to” certain special Uα’s. Specifically, for each set B ∈ B, we will select one set UB from among the Uα’s as follows: First ask if there exists at least one of the open sets Uα containing that set B. If not, let UB = U0 = ∅. If the set B is contained in some Uα, then pick one such Uα and call it UB. We might pick the same Uα corresponding to several B’s (because a given Uα usually contains many basis sets), but we have at most one Uα chosen for each B; so the set {UB : B ∈ B} is countable. We now show that {UB : B ∈ B} covers X. Let x ∈ X. We shall prove that at least one of the sets UB contains x. Since the Uα’s cover X, there is some Uα containing x. Since B is a basis, there exists B ∈ B with x ∈ B ∈ Uα. Since that basis set B is contained in some Uα, B is one of the basis sets for which we chose a set UB ⊇ B. So x ∈ B ⊆ UB, in particular x ∈ UB.  The previous theorem says that having a countable basis for the topology is the strongest of the countability properties. The next example shows that it is strictly stronger, that is the other properties do not imply it. Example (ex 3, page 192). The space Rℓ is first-countable, separable, and Lindelöf , but not second-countable. Proof. The details are given in the text; you should be able to prove Rℓ is first-countable and separable, and that it is not second-countable. You are not required to know the proof that Rℓ is Lindelöf .  The previous example shows some of the independence of the properties. However, in metric spaces, the first-countable, separable, and second-countable properties are equivalent. c©J. Simon, all rights reserved page 2