Math 5610 Fall 2017

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Course Information: Syllabus

Handout on ordinals.

Homework 1
1.) a.) Show that if $$(X,\tau)$$ is $$T_1$$ and $$\tau$$ is closed under arbitrary intersections, then $$\tau$$ is the discrete topology on $$X$$.
b.) Let $$X=(0,1) \cup (2,3)$$ and let $$\tau$$ be the following collection: $$U \in \tau$$ iff $$[(U \cap (2,3) \neq \emptyset) \to ( (0,1) \subseteq U)]$$. Show that $$\tau$$ is $$T_0$$ but not $$T_1$$, and $$\tau$$ is closed under arbitrary intersections.

2.) Let $$X,\tau$$ be a topological space. Define a binary relation $$\sim$$ on $$X$$ by: $$x \sim y$$ iff $$\forall U\in \tau\ (x \in U \leftrightarrow y\in U)$$.
a.) Show that $$\sim$$ is an equivalence relation on $$X$$.
b.) Let $$Y=X/\sim$$ be the set of equivalence classes of $$X$$ with respect to $$\sim$$. Define $$\sigma=\{ V \subseteq Y\colon \cup V \in \tau \}$$. Here $$\cup V$$ denotes the union of the set $$V$$, that is the set of all elements of all the equivalence classes in $$V$$. Show that $$\sigma$$ is a topology on $$Y$$ and that it is $$T_0$$.

3.) Show that if $$(X,\tau)$$ is $$T_1$$ and $$A \subseteq X$$, then $$A'$$ is closed and $$A''\subseteq A'$$.

Notes on metric spaces: metric spaces, TeX file