Section3.5Separation and Countable Basis Properties
“Separation is not the end of love; it creates love.”
―Nancy Friday
Motivation: Characterize metrizability (in particular for compact spaces) in terms of properties defined via open sets.
Subsection3.5.1Countable Basis
Definition3.5.1.
A space \((X,\cT_X)\) has a countable basis if the topology \(\cT_X\) is generated by a basis that is a countable set.
Theorem3.5.2.
A countable basis is a homeomorphism invariant.
Theorem3.5.3.
The property of having a topology generated by a countable basis is preserved by subspaces and countable products.
Example3.5.4.
Examples
Theorem3.5.5.
If \((X,\cT_X)\) is a compact and metrizable space, then \(\cT_X\) is generated by a countable basis.
Metrizability does not imply a countable basis.
Subsection3.5.2Separation properties
Definition3.5.6.
Let \(X\) be a topological space.
\(X\) is \(T_1\) if for any points \(a,b \in X\) with \(a \ne b\text{,}\) there are open sets \(U,V\) in \(X\) with \(a \in U\text{,}\)\(b \not\in U\text{,}\)\(a \not\in V\text{,}\) and \(b \in V\text{.}\)
\(X\) is \(T_2\text{,}\) or Hausdorff, if for any points \(a,b \in X\) with \(a \ne b\text{,}\) there are open sets \(U,V\) in \(X\) with \(U \cap V = \es\text{,}\)\(a \in U\text{,}\) and \(b \in V\text{.}\)
\(X\) is \(T_3\) if \(X\) is \(T_1\) and for any point \(a \in X\) and closed set \(B\) in \(X\) with \(a \not\in B\text{,}\) there are open sets \(U,V\) in \(X\) with \(U \cap V = \es, a \in U\text{,}\) and \(B \sse V\text{.}\)
\(X\) is \(T_4\) if \(X\) is \(T_1\) and for any closed sets \(A,B\) in \(X\) with \(A \cap B = \es\text{,}\) there are open sets \(U,V\) in \(X\) with \(U \cap V = \es\text{,}\)\(A \sse U\text{,}\) and \(B \sse V\text{.}\)
Remark3.5.7.
If \(X\) is a \(T_1\) space, then \(T_3\) is also called regular and \(T_4\) is also called normal.
Theorem3.5.8.
For each \(1 \leq i \leq 4\text{,}\) the property \(T_i\) is a homeomorphism invariant.
Proposition3.5.9.
A space \(X\) is \(T_1\) if and only if all one-points sets are closed in \(X\text{.}\)
Theorem3.5.10.
For each \(2 \leq i \leq 4\text{,}\) the property \(T_i\) implies the property \(T_{i-1}\text{,}\) but the converse fails.
Example3.5.11.
Examples
Interactions with constructions.
Theorem3.5.12.
\(T_2\) and \(T_3\) are preserved by subspaces and products.
Remark3.5.13.
\(T_2\) and \(T_3\) are not preserved by quotients. \(T_4\) is not preserved by subspaces, products, or quotients.
Theorem3.5.14.
A space \(X\) is \(T_3\) iff \(X\) is \(T_1\) and for all \(x \in X\) and open \(U\) in \(X\) with \(x \in U\text{,}\) there is an open set \(V\) in \(X\) with \(x \in V \sse \text{Cl}_X(V) \sse U\text{.}\)
Theorem3.5.15.
A space \(X\) is \(T_4\) iff \(X\) is \(T_1\) and for all closed \(A\) and open \(U\) in \(X\) with \(A \sse U\text{,}\) there is an open set \(V\) in \(X\) with \(A \sse V \sse \text{Cl}_X(V) \sse U\text{.}\)
Subsection3.5.3Interactions among homeomorphism invariants
Theorem3.5.16.
Metrizable spaces are \(T_4\text{.}\) (The converse is not true.)
Theorem3.5.17.
Compact Hausdorff spaces are \(T_4\text{.}\) (The converse is not true.)
Theorem3.5.18.
A \(T_3\) space with a countable basis is \(T_4\text{.}\)
Theorem3.5.19.Urysohn's Lemma.
Let \(X\) be a \(T_1\) space. The following are equivalent:
\(X\) is \(T_4\text{.}\)
Whenever \(A\) and \(B\) are disjoint closed subsets of \(X\text{,}\) there is a continuous function \(f : X \to [0,1]\) such that \(f(A) =\{0\}\) and \(f(B) =\{1\}\text{.}\)
Theorem3.5.20.UMT = Urysohn Metrization Theorem.
If \(X\) is a \(T_3\) space with a countable basis, then \(X\) is metrizable.
Corollary3.5.21.
Let \(X\) be compact. The following are equivalent:
