Let $R$ be a closed equivalence relation on a Hausdorff space $X$. Show that $X/R$ is Hausdorff.
14282
$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$
149820
How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
46698
Showing a set is closed under inverses
20736
Prove the relation $R$ in $N \times N$ defined by $(a,b) \simeq (c,d)$ iff $ad=bc$ is an equivalence relation.
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
46287
Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
2575
Showing that $\mathbb{R}$ is connected
116009
Proving that if a set is both open and closed then it is equal to the real numbers
55902
Show $R[x]/(x)\cong R$
80209
If $X,Y$ are equivalence relations, so is $X \times Y$
70049
Irreducible topological spaces and irreducible Hausdorff space
3896
A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
98630
For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation.
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
118851
$X$ be a Hausdorff topological space , let $f:X \to X$ be a continuous map such that $f\circ f=f$ , then is $f(X)$ is closed in $X$ ?
18034
Is there an equivalence relation of which the quotient topology is homeomorphic to codomain of a quotient map?
138202
Every open set in $\mathbb{R}$ is a countable union of closed sets