(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 3AM 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. Thenql_b80996293042b124409b5f6598f26b2e_l3Therefore this shows that \bigcup U_\alpha\in\mathcal{T}_C for all \alpha\in J.
  3. And fl_1ff4a3497a229f47039b7f9587eacc54_l3Therefore 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

 

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