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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Is the colimit of an expanding sequence of $T_4$ spaces $T_4$?

Let $$X_1\subseteq X_2\subseteq\dots\subseteq X_n\subseteq\dots$$ be an expanding sequence of spaces. Write $X_\infty$ for the colimit of the sequence. i.e. $X_\infty=\bigcup X_n$ topologised so that ...
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CW complexes are T$_1$

Given the constructive definition of CW-complexes (i.e. the one Hatcher gives in his Algebraic Topology book) how would one prove that every singleton in closed. He states in page 522 that every point ...
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Proof of Equivalent Characterisation of Complete Metric Space without using Compactifications

While studying about Completely Metrizable spaces, I came across this theorem - For a metric space $X$, the following are equivalent $X$ is completely metrizable $X$ is a $G_\delta$ in its ...
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Necessary and Sufficient conditions for Product of Limit Point Compact spaces to be Limit Point Compact?

Do there exist necessary and sufficient conditions for the product of Limit Point Compact spaces to be Limit Point Compact? I've been able to find some conditions for this to hold for Countably ...
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Minimal Conditions for Limit Point to be an $\omega$-Accumulation Point?

Let $X$ be a topological space. I know that if $X$ is $T_1$, then every limit point of $Y$ is an $\omega$-accumulation point, and if $X$ is $T_0$, then this does not hold. So, are there any separation ...
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Convergent Sequences of Extremally Disconnected Hausdorff Spaces

It's written in Willard (15G.3) that the only convergent sequences in a Hausdorff Extremally Disconnected space are the eventually constant sequences. However, it has not provided a proof. I've tried ...
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Dense Subspace of Extremally Disconnected Space is Extremally Disconnected

Problem 15G of Willard is - Every dense subspace and every open subspace of an extremally disconnected space is extremally disconnected. I've been able to prove the 'open subspace' part of the ...
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Embedding Regular Lindelöf Space in a Hausdorff Space

Preliminary definition - $G_{\delta}$ Closed: A set $A$ is $G_{\delta}$ closed if each point $x\not\in A$ is contained in a $G_{\delta}$ set disjoint from $A$. Willard states that - A regular space ...
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Example of Separable Product Space with cardinality greater than continuum?

In Willard, it's given that, for Hausdorff non-singleton spaces - $\prod_{\alpha\in A}X_\alpha$ is separable iff $X_\alpha$ is separable $\forall\alpha\in A$ and $|A|\le\mathfrak{c}$ From reading ...
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Example of $\text{Regular}$ non-$T_0$ space which is not $\text{Completely Regular}$?

Note: I'm assuming that the definition of Regular and Completely Regular spaces do not require them to be $T_0$. While studying Topology, I've found examples that show that $\text{T}_3 \nRightarrow \...
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Are Semiregularity and Complete Hausdorff Properties preserved by Products

By preserved by products I mean - $\prod X_{\alpha}$ has property $P$ iff $X_{\alpha}$ has property $P$ for all $\alpha$ in index set Also, $X$ is Completely Hausdorff if for $x\neq y$ in $X$, $\...
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Example of Semiregular and $T_1$ Space which is not Hausdorff

Here, my definition of Semiregular Space is $-$ a space which has a base of regularly open sets. In Willard, it is written that - A Semiregular $T_1$ space need not be $T_2$ However, Willard has ...
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Existence of Hausdorff space comprised of disjoint dense subsets

Problem $13H$ of Willard states - For any set $X$, there is a Hausdorff space $Y$ which is the union of a collection $\{Y_x:x\in X\}$ of disjoint subsets dense in $Y$. I have no idea how to do this. ...
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Hausdorff topology on manifolds

I've to prove the following result but I'm not sure about my solution. Prove that the topology on a manifold $M$ is of Hausdorff iff it admits an atlas $\mathcal{A}=\{U_{\alpha}, \phi_{\alpha}\}$ with ...
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Separation axioms about precisely separated by continuous function

Do the two axioms that "distinct points/closed set and point outside are precisely separated by continuous function" have their names? (Compared to perfectly normal, maybe we should call ...
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Separation axiom implied by semidecidability of comparison

I am studying computable analysis. What I'm fascinated by is the analogy between computable analysis and general topology: a Wikipedia article Semidecidable sets are analogous to open sets. So I ...
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Convergent sequence and Hausdorff space

It's trivial that any sequence in a Hausdorff space converges to at most one point. What about its inverse? E.g. If a topological space have the property that any sequence in it converges to at most ...
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Are T1 and T0 inheritable in the subspace and product topologies?

T2-ness property is inherited in subspaces and in the topological product. What about T1 and T0, are they inheritable? Can someone give simple examples when this happens and when this does not happen, ...
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$X$ is a $T_1$ space but $X/$~ is not $T_0$: an example of a such space $X$?

I'm looking for a $T_1$-space (or a $T_0$-space) and an equivalence relation ~ on $X$ such that the resulting quotient space $X/$~ is not $T_0$. My idea: take $X=\mathbb{R}$ with the usual euclidean ...
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Does every non-compact Tychonoff space admit an unbounded continuous function? [duplicate]

Let $X$ be a completely regular Hausdorff space. Such a space is also known as Tychonoff space, or a $T_{3.5}$-space. Furthermore, let's assume that $X$ is not compact. Question. Does $X$ admit a ...
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A sufficient condition for a space not to be $T_1$ by a collection generating the topology.

I've thought about the following result, which I wanted to verify: Let $X$ be a topological space where $\vert X\vert>1$ generated by a collection of subsets $\{ S_\alpha \}_{\alpha\in \Lambda}$, ...
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Is a continuous image of a normal space normal?

Problem Let $f:X\rightarrow Y$ be closed continuous surjective map. Show that if $X$ is normal then So is $Y$. What if we drop the 'closed' condition? I want a counter example. I know the proof of ...
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Proof Verification: Equivalent Definition for Locally Compact Hausdorff Space

The main theorem is as follows. I think most people are familiar with that: Theorem. Let $X$ be a Hausdorff space. Then $X$ is locally compact if and only if for every $x\in X$ and every open set $U$ ...
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Separation argument and polar cone

Let $S\subset \mathbb{R^n}$, $T\subset \mathbb{R^n}$ be nonempty closed convex cones. Define $C^*=\{ y\in \mathbb{R^n} | y\cdot x \leq 0, \forall x \in C\}$ I am trying to show $(S\cap T)^* \subset ...
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Lower limit topology on $\mathbb R$ is regular

I want to prove that lower limit topology $\Bbb R_l$ is regular and am taking the approach as follows: Let $A$ be a closed set and $x$ a point in $\Bbb R_l$ such that $x \notin A$. Let $a \in A$. If $...
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Extension theorems in a topological space

I have to prove some cases of extention problems one of them is : If $X$ be a completely regular space and $E$ is a subspace of $X$ which consists of a closed set $F$ of $X$ and a point $p \in X \...
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Prob. 7 (b), Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a regular space under a perfect map is also a regular space

Here is Prob. 7 (b), Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is ...
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Prob. 7 (a), Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a Hausdorff space under a perfect map is also a Hausdorff space

Here is Prob. 7 (a), Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is ...
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If a homotopy can be extended to a neighborhood of a closed subspace of a normal space $X$, then it can be extended to all of $X$.

During some self-study, I came across the following problem in Spanier's Algebraic Topology: Statement: Suppose $X$ is a normal space, and $A$ is a closed subspace of $X$. Let $f\colon X \to Y$ be a ...
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Prob. 4, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Satisfaction or otherwise of separation axioms as topology on a set become coarser or finer

Here is Prob. 4, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $X$ and $X^\prime$ denote a single set under two topologies $\mathscr{T}$ and $\mathscr{T}^\prime$, respectively;...
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Verifying separation axioms for the product topology $\{0,1\}^J$

Is my proof efficient? I think I was able to verify the separation axioms but I am still not entirely sure. Any help is greatly appreciated! Thanks! $\def\R{{\mathbb R}}$ I wish to prove the ...
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$T_0$-Identification of a topological space

In Willard's General Topology, section 13.2.c, for any topological space $X$ is defined a quotient space $X/\sim$ such that $x \sim y$ iff $cl(\{x\})=cl(\{y\})$ where $cl(.)$ is the topological ...
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Additional consequence of Urysohn's Lemma?

Urysohn's Lemma states that for any normal topological space $X$ and closed disjoint subsets $A,B\subset X$, we can find a continuous function $f\colon X\to[0,1]$ such that $f|_{A}=0$ and $f|_{B}=1$. ...
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Separated set in subspace topology.

If $A,B\subset E\subset X$ where $X$ is a topological space.Then is the following true? $A,B$ are separated in $X$ $\iff$ $A,B$ are separated in $E$. i.e. $\overline{A}\cap B=A\cap \overline{B}=\phi$...
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Is there typographical error in Stephen Willard's General Topology proof of Theorem 28.11

Here is 28.11: The proof on page 206 initially refers to separation order E(a,b). In the second paragraph, it supposes distinct points in E(a,c) - {a,b}. I am reading this on my own so I have no-one ...
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Questions about completely normal spaces.

I'm trying to solve the next problem: A topological space $(X,\tau)$ is called completely normal if, and only if, every subspace is normal. Prove that the following conditions are equivalent: a) $X$ ...
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Proof: Every point-finite open cover in a normal space has an open shrinkage

I try to unterstand the proof of the statement: Each point-finite open cover $\mathcal{U}$ of a normal topological space $(X,\mathcal{T})$ has an open refinement $\{V_{U}\mid U\in\mathcal{U}\}$ ...
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Can we find $\lambda$ with $B(\lambda, |\lambda|) \subseteq K?$

Let $K$ be a compact subset of the positive real axis, viewed as a subset of the complex plane $\mathbb{C}$. Let $a$ be an element not on the positive real axis, i.e. $a \in \mathbb{C}\setminus [0, \...
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Cardinals comparison in Jones' lemma

In Willard's General Topology, (ed. Dover, p100), the lemma 15.2 (Jones' lemma) states that: If $X$ contains a dense set $D$ and a closed, relatively discrete subspace $S$ with $|S| \ge 2^{|D|}$, ...
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$T_1$-property of cofinite topology.

Define $\tau$ on $\mathbb N$ by $\tau:=\{A\subset \mathbb N:\mathbb N-A $ is finite$\}(=\tau_f)$.Now I want to show that $\tau_f$ gives rise to a $T_1$ space on $\mathbb N$ although it is not $T_2$....
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How to simply represent topological spaces satisfying separation axioms.

I am a new learner of topology and I feel confused when I am introduced to different separability axioms like $T_0,T_1,T_2$ etc.Is there any way to diagramatically represent these spaces by simple ...
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Hausdorffness on Topological Quotient Group

Let $G$ is a topological group i.e. operation of the group and inverse are continuous wrt topology. I should show that if $G$ is Hausdorff then $G/N$ is Hausdorff where $N$ is a normal subgroup of $G$...
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Deducing two points are topologically indistinguishable by sub-base

I have the following argument which I would like to verify: Let $X$ be a topological space with a sub-base $\{ S_\alpha \}_\alpha$ such that $\cup S_\alpha\neq X$. Then every $x,y\in X\setminus \...
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Checking separation axioms against topological sub-basis

This is in sorts a follow up to the question Checking separation axioms against topological basis, in which I verified that checking certain separation axioms can be done only with Basis elements. My ...
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Checking separation axioms against topological basis

I've been interested lately in topologies whose definitions are given by generating sets, and I was trying to ascertain a few properties of theirs. However since I am unsure how a general open set ...
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$T_4$-ness that is preserved by product

Sorgenfrey line demonstrates how normality can be not preserved when "squared." Is there an example for a normal space $X$ for each of?: $X^2$ is normal, but $X^3$ is not $X^2$ and $X^3$ are normal, ...
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can we “represent” this finite topology as submodules of $R^X$?

Specific Question Let $X:= \{a,b,c,d\}$ equipped with the topology with base $c,d,abd,abc$ – where out of convenience we identify $x$ with $\{x\}$ and leave out union signs. Is there a ring $R$ ...
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not Hausdorff, question

I am trying to understand an example for a space which is not Hausdorff. I do not really see, why (Q1******) and (Q2******)marked underneath hold. To show: $Y$ (see definition underneath) is not ...
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Compact Hausdorff space and closure [closed]

Let $X$ be a compact Hausdorff space, let $A \subset X$ and let $U$ is open subset of $X$ such that $\overline A \subset U$. prove that there exist an open set $W$ such that $\overline A \subset W$ ...
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Example of regular space for which exist two distinct points that can not separate by continuous function

In the book "General topology" by Engelking, where is exercise to build regular topological space, for which there exist two distinct points, $a$ and $b$, for which is correct sentence: for every real ...

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