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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Let $X$ be a Hausdorff space and $A \subset X$ compact. Show that $A$ is closed.

Let $X$ be a Hausdorff space and $A \subset X$ compact. Show that $A$ is closed. Pick $x \in X \setminus A$, then since $X$ is Hausdorff there exists disjoint $O_x$ and $O_A$. Since $O_x \subset X \...
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Perfect map and one- point compactification

Let X and Y be non-compact locally compact $ T_{2} $-spaces and f : X → Y continuous map. Define $ f^{*} $ : σ(X) → σ(Y) be a map with : • $ \ f^{*} |{x} $ = f • $ f^{*} $($ \infty_{X} $) = $ \...
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proof of a separation result using the Hahn-Banach theorem

We work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to prove the subsequent claim using the Hahn-Banach separation theorem (Theorem 2.1 in this paper): Claim. Fix conjugate $...
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Let $X,Y$ be Hausdorff and $f,g : X \to Y$ continuous. Let $A \subset X$ be dense and suppose that $f\mid_A = g\mid_A$. Show that $f=g$.

Let $X,Y$ be Hausdorff and $f,g : X \to Y$ continuous. Let $A \subset X$ be dense and suppose that $f\mid_A = g\mid_A$. Show that $f=g$. Suppose the contrary that $f \ne g$. This implies that there ...
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Does $T_3$ imply that the topological space is zero-dimensional?

First, let me clear up my definitions: A topological space is $T_3$ if, given any point $x$ and closed set $F$ in $X$ such that $x$ does not belong to $F$, they are separated by neighbourhoods. A ...
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Topology, Hausdorff and Fréchet space

Solving a problem I came to an apparent contradiction. The problem is the following: Let $X$ be a set and $p\in X$. We consider the following topology on $X$ $\tau_p = \{ A \subseteq X: p \notin A \} \...
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Let $U(a,t) = \{a\} \cup [t, \infty)$ where $a,t \in \Bbb R.$ Show that the generated space is not Hausdorff.

Let $U(a,t) = \{a\} \cup [t, \infty)$ where $a,t \in \Bbb R.$ The collection of these sets forms a basis for some topology $\tau$ on $\Bbb R$. Show that the generated space is not Hausdorff. Let $x,y ...
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Separation Axioms: is it true that $T_4 \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$?

I am thoroughly confused by the separation axioms in topology. My lectures, online resources, and books all seem to say different things. Online I read that $T_4 \Rightarrow T_3 \Rightarrow T_2 \...
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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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Study some topological properties of $I^{\aleph_0}\times I^2/M$

I've been solving some problems from my Topology course, and I'm unable to finish this one: Given $I^{\aleph_0}$ the Hilbert cube, and given $Y=I^2/M$ the quotient space where $M=\{(0,0),(1,0),(0,1),(...
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Standard name for this function $\mu_\mathcal{X}(a) = \underset{x \in \mathcal{X}}{\inf} a^Tx $

I'm working with expressions that utilize the support function of a set $\mathcal{X} \in \mathbb{R}^n$, i.e. $$\sigma_\mathcal{X}(a) = \underset{x \in \mathcal{X}}{\sup} a^Tx $$ where $a \in \mathbb{R}...
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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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Characterization of Hausdorff space

When we study Hausdorff space, we can have the following two results: Cor. 173 If $f,g: X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\mid f(x) = g(x)\}$ is closed in $X$. Cor. 174 If $f,...
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If $f$ is atomic then int $(f (U)) \neq \emptyset$ (interior). True or False?

Let $f:X \to Y$ be a continuous function between continua. If $f$ is atomic then int $(f (U)) \neq \emptyset$ (interior). I don't know if this conjecture is true. Before presenting my attempt, I ...
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$T=${$U \in P(\mathbb{R}): 0 \notin U$ or $ R-U $ is finite) is completely normal but not perfectly normal

This question is from Wayne Patty's Section 5.3 Let $T=${$U \in P(\mathbb{R}): 0 \notin U$ or $ \mathbb{R} \setminus U $ is finite } is (a) completely normal (b) not perfectly normal. (a) A $T_1$- ...
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A $T_1$ space is normal iff... [duplicate]

This question was given by my instructor to work by myself and I was unable to completely prove it. (a) Prove that a $T_1$ space (X,T) is normal iff for each closed subset C of X and each open set U ...
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Solution verification for a regular space

This question is from textbook Wayne Patty Exercise 5.2 Question 18. For each $(a,b)\in \mathbb{R}^2$ ; an open disk with center at (a,b) is a set of the form {$(x,y)\in \mathbb{R}^2 : (x-a)^2 +(y-b)^...
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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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Prove that this topological space is Haursdorff and Compact

The following question is from my topology assignment and I was unable to solve it. Consider the equivalence relation $\sim$ on $\mathbb{R} \times [0,1] $ defined by $(x,t) \sim (x+1, t)$, $x\in \...
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Complete regularity of function spaces

Let $X$ and $Y$ be topological spaces. Wikipedia states that if $Y$ is $T_{3½}$, then so is $Y^X$. Yet I couldn't find the proof anywhere. So here's my own attempt of a proof: (I also had had a hard ...
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Is the product of paracompact with $\sigma$-compact always paracompact?

Exercise A space $X$ is said to be $\sigma$-compact if it can be written as a countable union of compact subspaces. Is the product of a paracompact space $X$ with a $\sigma$-compact space $Y$ always ...
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Proving $\mathbb{R}$ is Hausdorff with final topology induced by a function $f$.

Consider a Hausdorff topological space $(X,\tau)$. Suppose $(X,\tilde{\tau})$ is the minimal normalization of $(X,{\tau})$, that is, for every given normal topology $\sigma$, where $\tau \subset \...
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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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K-topology satisfies the Hausdorff axiom

I want to prove that the topology on $\mathbb{R}_K$ satisfies the Hausdorff axiom. We know the topology on $\mathbb{R}_K$ is generated by basis $(a,b)$ and $(a,b)-K$ where $K =\{1/n\}_{n \in \mathbb{Z}...
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Reference request for Separation axioms in Topology.

I am an undergraduate student of Mathematics and I want to study the topic "Separation Axioms" of general topology.I have already studied Basis,Subbasis,Product topology,Countability axioms,...
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Necessity and sufficient condition of perfectly normality

In Engelking's book, there is exercise (p. 49, ex. 1.5.K), where is written that $T_1$ space $X$ is perfectly normal if and only if for every open set $W$ from $X$ where exist sequence $W_1,W_2,\ldots$...
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Non T1 space with every convergent sequence is eventually constant

I know that discrete space and cocountable space are $T1$ space with every convergent sequence is eventually constant (If we have a convergent sequence $(x_n),$ then $\exists\, n_0 \in N, \forall n\...
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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 ...
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why $U_{y_1}\cap ....\cap U_{y_n}$? why not $U_{y_1}\cup ....\cup U_{y_n}?$

I have some confusion about the statement in Munkras Book Theorem $26.3$: Every compact subspace of a hausdorff space is closed In the theorem of the proof it is written that the open set $V_{y_1} ...
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A $T_1$ - space is normal if and only if...

This question is from my topology quiz and I was unable to solve it. So, I am asking for help here. Prove that $T_1$ -space (X,T) is normal iff for each closed subset C of X and each open set U such ...
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Proving completely regular is a topological property

This question is from Wayne Patty's Topology Page 175. Prove that completely regular is a topological property. I chose C is closed in X and p$\in X-C$, there exists f such f(p)=1 and f(c)=0 for ...
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How do I show that if $A$ is compact and $U \supseteq A$ is open, then there is an open $V$ with $A \subseteq V \subseteq \overline{V} \subseteq U$? [duplicate]

This question is from Wayne Patty's Topology Section 5.2. Consider $A$ be a compact subset of a regular space and let $U$ be an open set such that $A\subseteq U$. Prove that there is an open set $V$ ...
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Collection of all compact subsets of a Hausdorff space $X$ is compact if and only if $X$ is compact.

Let $X$ be a Hausdorff space. Let $K(X)$ be the collection of all compact subsets of $X$. A topology on $K(X)$ is defined by a subbasis given by sets of the form $I_U=\{K\in K(X)\,|\,K\subset U\}$ and ...
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T1 space, there all closed G-delta set is zero-set, but isn't normal

In Engelking's General topology, in exercises part, there is Ju. M. Smirnov's charactarization of normal spaces: T1 space is normal iff following properties hold (both): a. every closed G-delta set is ...
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Space is $T_1$ iff finite subsets are closed: alternative approach

To go about proving the first direction $(\Rightarrow)$, we suppose $X$ is $T_1$. It then suffices to show that singletons are closed and then to rewrite a finite subset as the union of closed ...

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