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which of the following statements are true?

($a$) If every countable subset of a topological space is closed, then the space is discrete.
($b$) Every closed function from one space onto another is open.
($c$) Every discrete space is $0$-dimensional.


My thought :-

for a.I think it is false. but not sure. $\mathbb{N}$ with cofinite topology may work but not sure.
for b. let us consider $X$=[$-1,1$] and $f(x)=x^2$. so it is also false.
for c. I have no idea at all.

can anyone help me please

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1 Answer 1

up vote 2 down vote accepted

(a): You almost have the solution, just replace cofinite by cocountable and adjust the underlying set.

(b) Your example works, but only if the map is not surjective, that is if $Y\neq[0,1]$. For an example of a continuous and closed surjection, let $X=I$ and $Y=S^1$ and consider the map $x\mapsto(\cos(2\pi x),\sin(2\pi x))$.

(c) According to Wolfram MathWorld a space is $n$-dimensional if for every open cover there is an open refinment such that each point is in at most $n+1$ of the sets of the refinement. For a discrete space you can always take the discrete cover consisting of all the singletons.

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