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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\Delta_X$ ...
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### 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 ...
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### 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 ...
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 $U_1 \cap U_2 = \... 5answers 8k views ###$X/{\sim}$is Hausdorff if and only if$\sim$is closed in$X \times XX$is a Hausdorff space and$\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 \...
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 ?