List questions from separation-axioms | MathQA

For every closed set $F$ in a completely regular space $X,$ does there exist a nonzero continuous function $f:X\to [0,1]$ such that $f=0$ outside $F?$

Published on 15 Apr 2026 - 17:21

For a closed $G_\delta$ set $F\subseteq X,$ does there exist a continuous function $f:X\to [0,1]$ such that $f=0$ on $F$ and $f\neq 0$ outside $F?$

Published on 15 Apr 2026 - 22:43

Does there exist a continuous function separating these two sets $A$ and $B$

Published on 15 Apr 2026 - 10:38

What does it mean if you take the zero vector in the hyperplane separation theorem?

Published on 14 Apr 2026 - 6:44

Separating points in a compact Hausdorff space.

Published on 11 Apr 2026 - 21:47

Prime Integer Topology is $T_2$ but not $T_3$

Published on 13 Apr 2026 - 23:05

Locally compact Hausdorff then Tychonoff

Published on 11 Apr 2026 - 12:06

Prove that the function $f:X\to I=[0,1]$ by $f(x)=\min\{\frac{d(x,C)}{d(p,C)},1\}$ is continuous on $X$.

Published on 14 Apr 2026 - 17:34

Compact group acting on regular space

Published on 12 Apr 2026 - 2:13

How do I prove $U (A_1)\cap V (A_2) \neq \emptyset$? Can You help me to find where do we arrive contradiction?

Published on 11 Apr 2026 - 19:49

$T_0$-space iff for each pair of $a$ and $b$ distinct members of X, $\overline{\{a\}}\neq\overline{\{b\}}$

Published on 16 Apr 2026 - 11:53

Is a completely regular space whose convergent sequences are eventually constant discrete?

Published on 12 Apr 2026 - 2:54

How do I prove that there exists a bijection from power set of $D$ to $C'\subseteq C.$

Published on 09 Apr 2026 - 15:45

Nested interestection forms neighbourhood basis

Published on 10 May 2026 - 18:19

An equivalent condition for being $T_1-$space $(X,\mathscr T)$ is normal.

Published on 15 Apr 2026 - 20:14