A description/proof of reverse of Tietze extension theorem
Topics
About: topological space
About: map
The table of contents of this article
Starting Context
- The reader knows a definition of normal topological space.
- The reader knows a definition of continuous map.
- The reader admits the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous.
- The reader admits the proposition that the preimages of any disjoint subsets under any map are disjoint.
Target Context
- The reader will have a description and a proof of the reverse of the Tietze extension theorem: any topological space is normal if any continuous map from any closed set to \(\mathbb{R}\) has a continuous extension to the whole topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological space, \(T\), if any continuous map from any closed set to \(\mathbb{R}\), \(f: C \rightarrow \mathbb{R}\), has a continuous extension, \(\overline{f}: T \rightarrow \mathbb{R}, \overline{f}|_{C} = f\), \(T\) is normal.
2: Proof
Suppose that any continuous map from any closed set to \(\mathbb{R}\), \(f: C \rightarrow \mathbb{R}\), has a continuous extension, \(\overline{f}: T \rightarrow \mathbb{R}\). Let us think of any disjoint closed sets, \(C_1, C_2 \subseteq T, C_1 \cap C_2 = \emptyset\). Let us think of a continuous map, \(f: C_1 \cup C_2 \rightarrow \mathbb{R}\), such that for any \(p \in C_1\), \(f (p) = -1\); for any \(p \in C_2\), \(f (p) = 1\). \(f\) is continuous, because \(\{C_1, C_2\}\) is a closed cover of \(C_1 \cup C_2\) and \(f|_{C_1}\) and \(f|_{C_2}\) are continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous. There is a continuous extension, \(\overline{f}\). Let us define \(U_1 := {\overline{f}}^{-1} ( (-\infty, 0))\) and \(U_2 := {\overline{f}}^{-1} ( (0, \infty))\), open as preimages of open sets under a continuous map. \(U_1\) and \(U_2\) are disjoint by the proposition that the preimages of any disjoint subsets under any map are disjoint. \(C_1 \subseteq U_1\) and \(C_2 \subseteq U_2\). So, \(U_1\) and \(U_2\) are disjoint neighborhoods of \(C_1\) and \(C_2\). As \(C_1\) and \(C_2\) are arbitrary, any 2 disjoint closed sets have some disjoint neighborhoods, which is the definition of being normal.
