` Math 5620 Spring 2023

Math 5620 Spring 2023

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Course Information: Syllabus

1.) Let \( (X,\rho)\) be a metric space. Show directly that \(X\) is \(T_4\) as follows.
Let \(A,B\subseteq X\) be disjoint closed sets. Let \(f_A(x)= \rho(x,A)=\inf\{ \rho(x,a)\colon a\in A\}\), and likewise define \(f_B\). Show these are continuous functions, and then let \(F(x)= \frac{f_A(x)}{f_A(x)+f_B(x)}\).
Show that \( F\colon X\to [0,1]\) is well-defined, continuous, and \( F(A)=0\), \(F(B)=1\).

2.) Let \( X\) be a topological space, and let \( \mathcal{F}\) be the family of all real-valued continuous functions on \( X\).
Show that if \( X\) has the weak topology from the family \( \mathcal{F}\), then the family \( \mathcal{F}\) separates points from closed sets in \( X\).
[hint: Let \( U \subseteq X\) be open and \( x \in U\). Let \( I_1,\dots, I_n\) be open intervals in \( \mathbb{R}\) and \( f_1,\dots, f_n \in \mathcal{F}\) such that \( x \in f_1^{-1}(I_1)\cap \cdots \cap f_n^{-1}(I_n) \subseteq U\).
Get simple functions \( g_1,\dots, g_n \colon \mathbb{R}\to \mathbb{R}\) with \( g_i(f_i(x))=1\) and \( g_i=0\) off of \( I_i\). Combine the \( f_i, g_i\) into a single function \( F\colon X\to \mathbb{R}\).]