Questions tagged [separation-axioms]

Separation axioms are properties of topological space which, roughly speaking, say in what way two points, a point and a closed set, or two closed sets can be "separated". Most important are $T_0$-spaces, $T_1$-spaces, Hausdorff, regular, completely regular and normal spaces.

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$X$ is $T_1$ implies $f(X)$ is $T_1$

Let $X$ and $Y$ topological spaces, and $f:X \rightarrow Y$ a function such that $Im(f) = Y$ If $A\subset X$ is closed, so $f(A)$ is closed in $Y$. Let's suppose that $X$ is a $T_1$ space, is it ...
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Is the characterization of Hausdorff spaces in terms of ultrafilter convergence equivalent to the ultrafilter lemma?

It can be easily proven using the ultrafilter lemma that if every ultrafilter on a topological space converges to at most one point, then the space is Hausdorff. My question is is whether this ...
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Prove or disprove : In a topological space $(X,\tau)$ if every compact subsets $K\subset X$ are closed then $(X, \tau)$ is hausdorff.

$(X, \tau)$ be a topological space. $K\subset X$ is compact. I can prove if $X$ is hausdorff space then $K\subset X$ is closed. I know that the proof strongly requires the $T_2$- property of $X$. But ...
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Normal space on $[ -1,1]$

Consider the topology generated by the following base $$B = \{ [−1, b)\mid b > 0\} \cup \{(a, 1] \mid a < 0 \}$$ over $X = [−1, 1]$. Is the space $X$ a normal space? Edit: my solution is ...
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Prove that $A$ is closed.

let $f:X\to Y$ continuous, open and onto. Then $Y$ is Hausdorff if and only if the set $\{(x, y): f (x) = f (y) \}$ is closed. I already did the one for $Y$ is Hausdorff so the set is closed. ...
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Explain the argument used in the answer

the following post is the answer for Show that a locally compact Hausdorff space is regular. I was reading the answer,but not able get the highlighted argument. Please explain this... Suppose $X$ is ...
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Every locally Compact Hausdorff Space is Regular.

Every locally Compact Hausdorff Space is Regular $Proof$:Let $X=$locally compact+$T_2\implies X^*$ is compact+$T_2\implies X^*$ is Normal($T_4$)$\implies X^*$ is Regular$\implies X$ is regular($T_3$)...
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Why is paracompactness needed to prove a regular space is normal?

I am trying to understand why paracompactness is needed to prove a regular space is normal. It was used in the prove above to find a locally finite open refinment $\{w_\lambda\}$ from an open cover. ...
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Prove the proposition.

this is my first year studying topology and I was given a proposition but not its proof so I was wondering how i would make it but i didn't reach anything. It states the following: Let $X$ be any set ...
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Proof of normal topological space [duplicate]

$X$ is normal if for every two open sets $G_1,G_2\subseteq X$ with $G_1\cup G_2=X$, there exists closed sets $F_1\subseteq G_1$ and $F_2\subseteq G_2$ which also satisfies $F_1\cup F_2=X$. Where can I ...
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If every bijective map $M\to M$ is a homeomorphism, when is $(M, \tau)$ discrete?

Let $(M, \tau)$ be a topological space with the following property: (B) Every bijective map $f: M\to M$ is a homeomorphism. Under what extra assumption(s) can we conclude that $\tau$ is discrete? ...
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About butterfly points and non normality.

I'm reading this article and others about the same topic and I found that all autors assume the next theorem. First, a definition. Here $Y^{*}:=\beta{Y}\setminus Y$, i.e., $Y^{*}$ is the remainder of ...
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Every hausdorff space has a non-hausdorff quotient.

I know some Hausdorff spaces can have non-hausdorff quotient spaces. For example real line with double origin. I am wondering whether this is the case for all Hausdorff spaces. I tried some simpler ...
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A $T_{3\frac{1}{2}}$ space is compact if Stone-Weierstrass theorem holds.

A $T_{3\frac{1}{2}}$ space is compact if Stone-Weierstrass theorem holds. One proof of this is illustrated in Hewitt's "Certain generalizations of the Weierstrass approximation theorem". A ...
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$T_1$ space violating cardinal bound which every $T_2$ space satisfy

Let $X$ be a $T_2$ space. $T_2$ space is a topological space where every singleton is the intersection of its closed neighbourhoods i.e. $\{x\} = \bigcap_{x \in U, U \text{open}} \overline{U}$. If $D$ ...
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How can a regular space not be a Hausdorff space?

How can you have a non-regular Hausdorff space? Both definitions seem to be identical, 2 points with disjoint neighborhoods. Pls don't point out that this is not a formal definition. It is a simple (...
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Does compact normal space implies metric space?

I know compact Hausdorff space is normal, and metric space is normal. What about compact normal space? Is it a metric space? My naive guess is that normality is nothing to do with metric and ...
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Open mapping from normal space to T1 space

There is an exercise in Engelking`s "General topology" (p. 49, ex. 1.5.M), to give an example of an open surjective mapping of a normal space onto a T1 space that is not a T2. Please, help ...
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Is this topological space normal?

Let $(X,\leq)$ a totally ordered set with, at least two elements. Let $\mathcal B = \{B_x\mid x\in X\}$ with $B_x=\{y\in X\mid x\leq y\}$ and $T$ the topology in $X$ generates by $\mathcal B$. I have ...
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Is $(\mathbb{R}, \tau)$ normal, irrespective of the nature of $\tau$?
Is a topological space $(\mathbb{R}, \tau)$ normal, irrespective of the nature of $\tau$? For convenience I post the definition of normality: Definition A topological space $(X, \tau)$ is said to be ...