## (DISCONTINUED) #1: Munkres Problem Set 2.13.1, 2.13.3

As an introductory remark for this being my first set of problems from the Munkres text, I have a few things to note.

• The numbering system corresponds as follows: “<chapter>.<section>.<problem number>”.
• Please feel free to correct any mistakes that I make in the comments and I will update them, as I am doing this for my benefit and the readers benefit as well.
• The schedule for these posts will solidify for every Monday at $3$AM PST.
• You can find Munkres’s Topology in the References section on my blog if you need a quick link to the text at your convenience.

For this problem set, I am sorry it has only two problems. I will try to post a few more following up in the middle of the week so that can everyone see more. If not, we will start fresh next week.

2.13.1

Problem: Let $X$ be a topological space; let $A$ be a subset of $X$. Suppose that for each $x\in A$ there is an open set $U$ containing $x$ such that $U\subset A$. Show that $A$ is open in $X$.

Answer: Let $U_x$ denote the open set where $x\in U_x\subset A$. For each $x\in A$, there will be an open set $U_x$ associated with it. Therefore $\bigcup_{x\in A}U_x=A$. By the definition of an open set, $A$ is an open set by the fact that the union of opens sets forms an open set. $\blacksquare$

2.13.3

Problem: Show that the collection $\mathcal{T}_C$ given in Example $4$ of Section $12$ is a topology on set $X$. Is the collection

$\mathcal{T}_\infty=\{U|X-U$ is infinite or empty or all of $X\}$

a topology on $X$?

Answer: By the definition of a topology,

1. $X-X=\emptyset$ and $X-\emptyset=X$; therefore $X$ and $\emptyset$ are in $\mathcal{T}_C$.
2. Let $\{U_\alpha\}_{\alpha\in J}$ be an indexed family on nonempty elements of $\mathcal{T}_C$. ThenTherefore this shows that $\bigcup U_\alpha\in\mathcal{T}_C$ for all $\alpha\in J$.
3. And Therefore this shows that $\bigcap_{i=1}^n U_i\in\mathcal{T}_C$ for some $i$.

Thus $\mathcal{T}_C$ is a topology on $X$. $\blacksquare$