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The questions are the following:

Consider the five topologies on the real line $\mathbb R$:

  1. $\mathcal T_1$: the standard topology

  2. $\mathcal T_2$: the $K$-topology

  3. $\mathcal T_3$: the finite complement topology

  4. $\mathcal T_4$: the upper limit topology

  5. $\mathcal T_5$: the topology generated by the basis $\{(-\infty,a)\mid a \in \mathbb R\}$

Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of these topologies.

My answer is the following:

$\mathcal T_1$: $\mathrm{cl}(K)=\{0\} \cup K$.

$\mathcal T_4$: $\{0\} \cup K$.

$\mathcal T_5$: $[0, \infty)$

Thank you.

  • 10
    Which can you solve? What have you tried for the others, and what problems have you met?2011-10-23
  • 7
    math.stackexchange.com is not a black-box. Where do you get stuck? Can you see when a set is closed in the different topologies?2011-10-23
  • 2
    What they said. The purpose of this problem is to help you learn the definitions. So try to do them yourself. If you want, write your solutions here for comments.2011-10-23
  • 4
    You should also explain what the $K$-topology is. (The others are standard.)2011-10-23
  • 0
    Note that in the $K$-topology your set by definition is closed, so what is its closure? Also, in the finite complement topology on an infinite set, every infinite set has the whole space as its closure. This is pretty easy as an exercise: every non-empty open set can only miss finitely many points, so cannot miss the infinite set.2011-11-23

2 Answers 2