# Thread: Finding the closure of sets in a given topology

1. ## Finding the closure of sets in a given topology

Hi guys,

I have to find the closure of $\displaystyle K = \left\{ \frac{1}{n} : n\in\mathbb{N} \right\}$, and $\displaystyle A=(0,1)$ in various topologies.

1. In the standard topology, I think $\displaystyle \overline{K} = K \cup \{0\}$, and $\displaystyle \overline{A} =[0,1]$

2. Now, for the finite complement topology, I'm unsure for both K and A.

3. The upper limit topology, I'm also unsure for both K and A.

4. The topology with basis $\displaystyle \left\{(-\infty, a) : a\in\mathbb{R} \right\}$ I think for both K and A is the set of all positive real numbers.

Thanks alot in advance,

HTale.

2. ## Closure

1. You are correct.

2. Take any point x in R and fix an open set U containing it. Since the complement of U is finite it contains at least one member of each of K and A. Hence the closure of K is the whole of R, as is the closure of A.

3. Not quite sure what you mean by upper limit topology, but I think it is the topology generated by open sets of the form (a,b]. On this assumption, the closures are, for K, K itself, and for A, (0,1].

4. This gives the set of all non-negative real numbers as the closure of both K and A.