** 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