Questions tagged [separation-axioms]

Let $X$ be a topological space. Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$. It is not hard to check that $P^X : \textbf{...

A subset $F$ if a topological space $X$ is called functionally closed if $F=f^{-1}(0)$ for some continuous map $f:X\to[0,1]$. It is clear that each functionally closed set $F$ in $X$ is a closed $G_\...

I was surprised to find out that, even if the normalization $X^\nu$ of a scheme $X$ is affine, $X$ may not be affine (remove the line $x=y$ from their example to make the source affine). In the ...

Let $U \subseteq X$ be an open and $Z := X \setminus U$ its closed complement. I want a sequence $$G_0(Z) \to G_0(X) \to G_0(U) \to 0.$$ However $X, U$ are not quasiseparated and perhaps not even ...