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Let $X=[-1,1]$ and $\textit{T}$=$\{A\in X|{0 \notin A}$ or $(-1,1)\subseteq A \}$. Are there sets from one element in the topological space $(X,T)$ such that they are: 1.closed, but not open, 2.open, but not closed 3. neither oper, nor closed 4.open and closed? I took the set $\{x\}$ for an $x\in X$ and different from $0$ and they all are open, except of ${0}$ who is closed and so there are no clopen sets, or sets that are neither open, nor closed. Am I judging it right?

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  • $\begingroup$ @BrianM.Scott Yes, that I meant, I corrected it. $\endgroup$ Apr 14 '21 at 19:58
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There are three kinds of singletons in this space:

  • $\{0\}$;
  • $\{1\}$ and $\{-1\}$; and
  • $\{x\}$ for $x\in(-1,1)$.

$X\setminus\{0\}$ is open, so $\{0\}$ is closed. $X\setminus\{1\}=[-1,1)$ and $X\setminus\{-1\}=(-1,1]$ are also open, since they contain $(-1,1)$, so $\{1\}$ and $\{-1\}$ are closed. If $x\in(-1,1)$, however, $\{x\}$ is not closed: in that case every open nbhd of $0$ contains $x$, so $0\in\operatorname{cl}\{x\}$.

Clearly $\{0\}$ is not open, and $\{x\}$ is open for $x\in X\setminus\{0\}$.

Thus,

  • $\{0\}$ is closed but not open;
  • $\{1\}$ and $\{-1\}$ are clopen; and
  • $\{x\}$ is open but not closed for each $x\in(-1,1)$.
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