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I was wondering if someone could help with the following problems:

(a) Give an example of a topological space $(X,\mathcal{T})$ and a subset $A$ of $X$ which is both open and closed.

(b) Give another example where $A$ is neither empty nor the whole of $X$.

(c) Give an example of a topological space $(X,\mathcal{T})$ and a subset $A$ of $X$ which is neither open nor closed.

My answers:

(a) Let $X$ be any set with any topology $\mathcal{T}$, let $A=\emptyset$.

(b) Let $X=(1,2)\cup(3,4)$, $A=(1,2)$ and $\mathcal{T}$ be the relative topology inherited from the usual topology on $\mathbb{R}$.

(c) Let $X=\mathbb{R}$ with the usual metric and let $U_{n}=(0,1+\frac{1}{n})$.Then take $A=\displaystyle\bigcap_{n=1}^{\infty} U_{n}=(0,1]$.

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  • $\begingroup$ Your answers are all correct. $\endgroup$ – user99914 Sep 14 '15 at 2:53
  • $\begingroup$ (b) discrete topology (c) consider a ball whose upper boundary is closed and lower boundary is open. And Oh there is an answer.. $\endgroup$ – Rubertos Sep 14 '15 at 2:53
  • $\begingroup$ As @John said, your answers are all correct. You’re working harder than necessary for (c), however, as you could simply set $A=(0,1]$ directly. $\endgroup$ – Brian M. Scott Sep 14 '15 at 2:54
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Your answers are totally correct. I fail to see, however, why you decided to write your answer for (c) as an infinite intersection.

To prove that $A = (0,1]$ is not open, it suffices to show that $1$ has no open neighborhood in $A$. To prove that $A$ is closed, it suffices to show that $0$ has no open neighborhood outside of $A$.

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  • $\begingroup$ I have to prove part (c) rigorously so I think it makes it slightly easier that way, Thanks for your quick response! $\endgroup$ – jackwo Sep 14 '15 at 2:56
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    $\begingroup$ Is there any quick way to see that such an infinite intersection is neither closed nor open? Perhaps you're seeing something that I'm not. $\endgroup$ – Omnomnomnom Sep 14 '15 at 2:57
  • $\begingroup$ @JohnMa good catch! Thanks. $\endgroup$ – Omnomnomnom Sep 14 '15 at 2:58

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