# Are there sets from one element in $(X,T)$ that are closed, but not open, open, but not closed, neither oper, nor closed, open and closed?

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?

• @BrianM.Scott Yes, that I meant, I corrected it. Apr 14 '21 at 19:58

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 $$x\in(-1,1)$$.