` 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 $$F\colon X\to \mathbb{R}$$.]