Linked Questions

2
votes
4answers
2k views

Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff [duplicate]

I think I have solved a problem in Topology by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight ...
5
votes
2answers
2k views

Show that $X$ is Hausdorff if and only if the diagonal $\Delta = \{(x, x):x \in X\}$ is closed in $X \times X$ [duplicate]

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify it or offer suggestions for improvement? Show that $X$ is Hausdorff if ...
1
vote
1answer
1k views

Closed set in a Hausdorff topological space [duplicate]

Possible Duplicate: $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed I'm trying to prove: If $X$ is a Hausdorff topological space and $\Delta \subset X\times X$ such that $\...
1
vote
1answer
350 views

A space $Y$ is Hausdorff iff the diagonal $D = \{(y_1,y_2)\in Y\times Y: y_1=y_2\}$ is closed [duplicate]

Show that the topological space $Y$ is Hausdorff iff the diagonal $D = \{(y_1,y_2)\in Y\times Y: y_1=y_2\}$ is a closed subset of $Y\times Y$. My attempt: First, suppose $Y$ is Hausdorff. Then $Y\...
0
votes
2answers
313 views

Prove that a topological space $(X, \tau)$ is $T_2$ if and only if the diagonal $D=\{(x,x):x \in X\}$ is closed subset of $X\times X$ [duplicate]

Prove that a topological space $(X, \tau)$ is $T_2$ if and only if the diagonal $D=\{(x,x):x \in X\}$ is closed subset of the product space $X\times X$ => assume that $(X, \tau)$ is $T_2$, I know ...
0
votes
3answers
204 views

Hausdorff and diagonal [duplicate]

Feedback on precision of argument and writing style is much needed. Suppose that $(X, \mathcal{T})$ is a topological space. Show that if $X$ is Hausdorff, the diagonal $\Delta = \{(x,x) \mid x \in ...
2
votes
1answer
382 views

A space is Hausdorff iff the diagonal is closed [duplicate]

Let X be a topological space. Prove that X is a T2-space if and only if $A = \{ (x,x) : x \in X \}$ is closed in $X \times X$. I've been able to prove first part of the result but not the converse. ...
-1
votes
1answer
150 views

$X$ is Hausdorff iff the diagonal is a closed subset of $X^2$ [duplicate]

Let $(X,\tau)$ be a topological space. Consider $X^2$ with the product topology. Show that $X$ is Hausdorff iff the diagonal $D = \{(x,y) \in X^2 \mid x=y\}$ is a closed subset of $X^2$.
1
vote
3answers
108 views

Prove that if $X$ is Hausdorff, $\Delta$ is closed in $X\times X$ [duplicate]

Prove that if $X$ is Hausdorff, $\Delta=\{(x, x)\mid x\in X\}$ is closed in $X\times X$ (with the product topology). My attempt: Let $x_1, x_2\in X$ s.t. $x_1\ne x_2$. There exist neighborhoods $...
1
vote
1answer
58 views

Prove $\Delta_X$ is closed iff X is Hausdorff [duplicate]

I'm attempting a topology proof and think my proof is correct but I'm not 100% sure. The problem is as follows: Define $\Delta_X=\{(x,x)\in X\}$. Prove that $X$ is Hausdorff if and only if $\Delta_X$ ...
1
vote
0answers
84 views

Proof: $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$. [duplicate]

Prove that $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$. This exercise appeared on a previous exam in my course, and also in Munkres. Here's my attempt: First I ...
0
votes
1answer
25 views

Whether a set is Hausdorff [duplicate]

To prove: A space $X$ is Hausdorff if and only if the set $\{(x,x)~|~x\in X\}$ is closed in $X\times X$. If $X$ is Hausdorff then if a sequence $\{x_n\}_{n=1}^{\infty}$ of elements of $X$ converges ...
1
vote
0answers
27 views

Question about Hausdorff spaces and their equivalences [duplicate]

Definition: A topological space $X$ is called Hausdorff space if for each $x_1,x_2 \in X$ (they are distinct) we can always find neighborhoods $U_1,U_2$ of $x_1,x_2$ such that $U_1 \cap U_2 = \...
29
votes
5answers
8k views

$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$

$X$ is a Hausdorff space and $\sim$ is an equivalence relation. If the quotient map is open, then $X/{\sim}$ is a Hausdorff space if and only if $\sim$ is a closed subset of the product space $X \...
6
votes
3answers
715 views

The set of points where two maps agree is closed?

Let $f,g\colon X \to Y$ be continuous maps. Let $Y$ be Hausdorff. Is the set $$A := \{x\in X \, : \, f(x)=g(x) \}$$ necessarily closed ?

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