Suppose that $X$ is a connected topological space with a connected subset $S$, and let $K$ denote a component of $X\setminus S$. (All sets listed are assumed non-empty, apart from $\emptyset$ of course.) Question. Is there an example of sets, as above, such that $\overline S\cap\overline K=\emptyset$? Remark. If there is such an example, then […]

Let $X$ be a topological space. Then, we say that $X$ is perfectly-$T_2$ or perfectly-hausdorff iff for every two distinct points $x_0,x_1\in X$, there is a continuous function $f\colon X\rightarrow [0,1]$ such that $\{ x_0\} =f^{-1}(0)$ and $\{ x_1\} =f^{-1}(1)$. That is, iff distinct points can be precisely separated by a continuous function. Is the […]

Let $X$ be a compact Hausdorff space such that for every $x \in X$ , there exist a nested sequence of open sets $\{U_n\}$ such that $\{x\}=\bigcap_{n=1}^\infty U_n$ , then is it true that $X$ is first countable?

This question already has an answer here: $f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$ 2 answers

A topological space $X$ is called locally Hausdorff if every point has a Hausdorff neighborhood. Local Hausdorffness is an interesting separation axiom that is strictly weaker than Hausdorff, but strictly stronger than $T_1$: The line with doubled origin is locally Hausdorff but not Hausdorff, and an infinite set with the cofinite topology is $T_1$ but […]

I have read that given any topological space $X$ you can construct a Hausdorff space $h(X)$ as a quotient of $X$ which is universal with respect to maps from $X$ to a Hausdorff space. This means there is a quotient map $q:X\to h(X)$ and if $f:X\to W$ is a map to a Hausdorff space, $W$, […]

It is not hard to show that if $f: X \rightarrow X$ is a continuous map and $X$ is a Hausdorff space, then the set of fixed points is closed in $X$. We basically just look at the diagonal and consider the map $g: X \rightarrow X \times X$ defined by $g(x)=(x,f(x))$. What happens if […]

A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets $E$ and $F$ can be precisely separated by a continuous function $f$ from $X$ to the […]

Let $X$ be a compact Hausdorff space and suppose the diagonal in $X\times X$ is the zero set of a nonnegative continuous real-valued function. Why does this imply $X$ is metrizable?

My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set (basic) $(a,b)$ such that $x \in (a,b)\subseteq X-F$. Then $(a,b)$ and $(-\infty,a) \cup (b,\infty)$ […]

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