--- 
title: "Local Neighborhood of a Quiz"
format: 
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            file: ../latex-header.tex
--- 

```{=latex}
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```

1. State the definitions of 
    1. a continuous function between topological spaces $X$ and $Y$.
       \vfill
    1. a closed set in a topological space $X$.
       \vfill
    1. limit point of a set $A$ in a topological space $X$.
       \vfill
    1. (local formulation of openness) a set $U$ is open if and only if ... (fill in the blank)
       \vfill
    1. Hausdorff space.
       \vfill
    1. a topological space being *discrete*.
       \vfill
    1. a basis for a topology on a set $X$.
       \vfill
1. True or False Section
    1. True or False: Any intersection of open sets is open.
    1. True or False: Any intersection of closed sets is closed. 
    1. True or False: Any union of closed sets is closed.
    1. True or False: Any union of open sets is open.
    1. True or False: A set $A$ is closed if and only if its complement $X \setminus A$ is open.
    1. True or False: A set $A$ is open if and only if its complement $X \setminus A$ is closed.
    1. True or False: A set $A$ is closed if and only if it contains all of its limit points.
    1. True or False: A set $A$ is open if and only if it contains all of its limit points.
1. Give an example of
    1. a continuous function $f : X \to Y$ between topological spaces $X$ and $Y$.
       \vfill
    1. a limit point of a set $A$ in a topological space $X$.
       \vfill
    1. a topological space that is Hausdorff.
       \vfill
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1. Short answer questions
    1. Let $X$ be a topological space and let $A \subseteq X$.  Prove that $\overline{A}$ is closed.
       \vfill
    1. Find the closure of the graph of the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. (Use Desmos if you like!) 
       \vfill
    1. Show the lower limit topology contains proper subsets that are both open and closed.
       \vfill
    1. Do continuous functions preserve limit points?  That is, if $f : X \to Y$ is continuous and $x$ is a limit point of a set $A \subseteq X$, is $f(x)$ a limit point of $f(A)$?  Prove or give a counterexample.
       \vfill
    1. Let $X$ be a topological space and let $A \subseteq X$.  Prove that $\overline{A} = A \cup A'$, where $A'$ is the set of limit points of $A$.
       \vfill
    1. Let $X$ be a topological space and let $A \subseteq X$.  Prove that $\overline{A}$ is the smallest closed set containing $A$.  That is, if $C$ is a closed set such that $A \subseteq C$, then $\overline{A} \subseteq C$. \vfill
    1. Is $\mathbb{Q}$ a closed set in $\mathbb{R}$?  Prove your answer. \vfill

