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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Compact space on ℤ

I want to make $ℤ$ (or equivalently, any countably infinite set) a compact space, as long as it satisfies a separation axiom. What is the maximum possible value of $x$ such that the space satisfies $...
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Is Mysior's example completely Hausdorff?

In the article Mysior, A., A regular space which is not completely regular, Proc. Am. Math. Soc. 81, 652-653 (1981). ZBL0451.54019. there is an example of the space that is regular, but not ...
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To prove that $X_{\infty}$ is normal

If $((X_n),(f_n))$ is a inverse limit system, and for each $n$, $X_n\neq {\emptyset}$ is compact and Hausdorff then $X_{\infty}$ is normal. I know there is a theory that says: If $((X_n),(f_n))$ an ...
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About Arens square. [duplicate]

In the famous book Counterexamples in topology by Steen and Seebach, is proved that Arens space is not a Urysohn space (in that book, a Urysohn space is the space in which any two distinct points can ...
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Separation axioms on a finite space imply discrete topology?

I saw somewhere on this site, a claim that the only topology on a mertizable finite space is the discrete one. I think this stems even more generally from such a space being Hausdorff. The strongest ...
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Questions about giving an example in subspace topology exercise

I just want to know if my guess for the following question is correct" Give an example of a separable Hausdorff space $(X,T)$ that has a subspace $(A, T_{A})$ is not separable. I am guessing $X=\...
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Question in the proof of Urysohn's Lemma

In the proof of the proposition, $X$ is normal $\implies$ $\forall E,F \subset X$ disjoint and closed $\exists f : X \rightarrow [0,1]$ continuous such that $\forall x \in E, f(x) = 0$ and $\forall x \...
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Altering affine space by changing offset within linear subspace of $\mathbb{R}^{n}$

Let $C\in\mathbb{R}^{n}$ be a convex set with $\text{aff}(C)=W+t$. For $x\in\text{cl}(C)\backslash\text{reint}(C)$, show that there exists $v\in W$, $v\neq0$, such that $$\text{sup}_{x\in C}v^{T}z=v^{...
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Restriction of Continuous Function on Compact Hausdorff Space [duplicate]

I'm trying to show that given a compact Hausdorff space $X$ and given a continuous function $f:X\to X$, there exists a compact subset $K\subseteq X$ such that $f_{\big|K}:K \rightarrow X$ maps $K$ ...
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When can we choose a covering local basis at a point which does not contain the whole space?

I was wondering if we can ensure by relatively mild seperation axioms that a local basis ,which is a cover of the space, for at a topology at a point $x_0\in X$ does not contain $X$? I'm pretty sure ...
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Proof verification: Metrizable spaces are completely regular.

Let X be a metrizable space with metric $d$. For disjoint closed sets $A$ and $B$ there is a continuous function $f:X\to[0,1]$ defined by $$f(x)=\frac{d(x,A)}{d(x,A)+d(x,B)}$$ such that $f(A)=\{0\}$ ...
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Inhomogenous Farkas’ lemma

Consider the following claim: Let $\mathbf{A}\in\mathbb{R}^{m\times n}$, $\mathbf{c}\in\mathbb{R}^{n}$, $\mathbf{b}\in\mathbb{R}^{m}$ and $d\in\mathbb{R}$. Suppose that there exists $0\le\mathbf{y}_{0}...
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Counterexamples regarding Totally Separated spaces

There are known the following implications regarding totally separated spaces: Every totally separated space is totally disconnected; Every totally separated space is Urysohn; Every zerodimensional ...
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When does trivial topology implies continuous functions are constant functions?

I found this question: $f$ is continuous if and only if $f$ is constant But I wondered whether all the assumptions are necessary. I think we can state the following claim as well: Let $f:X_1\...
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Is totally separated space equivalent to null dimensional and $T_0$?

It is a well known fact that every null dimensional $T_0$ space is totally separated. Is there an example of a totally separated space that is not null dimensional or $T_0$, or does the converse of ...
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Urysohn's lemma for $3$ closed sets?

$X$ is a normal Hausdorff space and $A,B,C$ three pairwise disjoint closed sets. I want to prove the existence of a continuous real-valued function $f$ that takes the values $a,b,c$ on $A,B,C$ ...
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normal and Hausdorff spaces

We say that a topological space $(X,\mathcal{T})$ is normal if any two disjoint closed subsets of $X$ are separated by neighbourhoods. We have that in $(\mathbb{R},\{\emptyset,\mathbb{R}\})$ are $$\...
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What qualifies as a “separation axiom?”

Wikipedia states the hierarchy of separation axioms as: $$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \...
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About perfectly normal spaces.

In some books the terms: regular and $T_3$; normal and $T_4$; completely normal and $T_5$; perfectly normal and $T_6$ are synonyms, but in some books, the difference is that regular, normal, ...
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Munkres Lemma 31.1, no need for $T_1$

In Munkres we have the following lemma: Lemma 31.1. Let $X$ be a topological space. Let one-point sets in $X$ be closed. (a) $X$ is regular ii and only if given a point $x$ of $X$ and a ...
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Does removing finitely many points from an open set yield an open set?

Removing finitely many point from an open set in $\mathbb{R}^n$ gives an open set. Is this true in general for any space? My intuition is that this is the case, however, how does one (dis)prove this?...
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How can we distinguish the two zeroes of these two double-line $T_1$-spaces?

Motivation If we have topological spaces which are not $T_1$, we can sort of measure the “failure to be $T_1$” by the specialization preorder $x\leq y$ iff $\operatorname{cl}(x)\subseteq\operatorname{...
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Hausdorff Space and Continuous map

Problem: Suppose $X$ is a topological space and for every $p\in X$ there exists a a continuous function $f: X\rightarrow \mathbb{R}$ such that $f^{-1}$ $\{$ $0$ $\}$ $=$ $\{$ $p$ $\}$. Show that $X$ ...
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What's an example of a non-Hausdorff $T_1$-space which is connected, but not hyperconnected?

The standard example of a non-Hausdorff space which is $T_1$ is either a cofinite space over an infinite base set or a cocountable space over something uncountable. However, those topologies are ...
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Why is T1-separability not preserved by continuous maps?

From Willard's "General Topology" (section 13.3), [edit: I thought that] it seems that the T1 separation axiom is preserved by the quotient topology, meaning that it is preserved by continuous maps. (...
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Compact-Hausdorff space in general topology

Why Is in general topology every Compact-Hausdorff space normal? Is every normal space Compact-Hausdorff space?
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$T_0$-identification of a topological space, proof using singleton closures

In Willard's General Topology, section 13.2.c, for any topological space X is defined a quotient space X/~ such that x ~ y iff $cl({\{x\}}) = cl(\{y\})$ where $cl(.)$ is the topological closure. Then ...
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Double derived set in $T_0$ spaces

Let $A$ be a subset of a topological space $X$. I am interested in establishing under which conditions the following inclusion holds: $A'' \subseteq A'.$ This is certainly false in general: consider ...
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How to show that filter convergence implies Hausdorff

Let $X$ be a topological space. Show that if every filter $F$ on $X$ converges to at most one point, then $X$ is Hausdorff. I want to argue by contrapositive. Suppose there exists a pair of points $x$...
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Classifying Hausdorff spaces

Can a topological space be Hausdorff and separable, but neither Lindelof nor first countable? Can a topological space be Hausdorff and Lindelof, but neither separable nor first countable? Can a ...
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Suppose a subspace of $X$ is compact iff closed. Then is $X$ compact hausdorff? [duplicate]

Let $X$ be a topological space. If $X$ is compact then any closed subspace is compact. If $X$ is hausdorff then any compact subspace is closed. So if $X$ is compact hausdorff, then a subspace is ...
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Proof that the spectrum of a ring is Kolmogorov

Let $A$ be a commutative ring and consider its spectrum $\operatorname{Spec}A$ equipped with the Zariski topology. Wikipedia claims that $\operatorname{Spec}A$ satisfies the separation axiom $\mathbf{...
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Any regular space isn't Hausdorff?

A space $X$ is regular if for all $x\in X$ and all closed $F\not\ni x$, there is $U,V$ open s.t. $x\in U$, $F\subset V$ and $U\cap V=\emptyset.$ In wikipedia, they talk about $T_3$ space as Regular ...
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Showing the normality of topological space

Let $f:\mathbb{R}\to(\mathbb{R},\mathcal{T})$ be a map defined by $f(x)=x$ if $x\in\mathbb{Q}$, $f(x)=0$ if $x\in\mathbb{R}\setminus\mathbb{Q}$, where $\mathcal{T}$ is the usual topology on $\mathbb{R}...
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Hausdorff and compact space

If $X$ is Hausdorff space and $x \in X$ isn't isolated point then $X-\{x\}$ isn't compact. Well, I am trying to show that if $X-\{x\}$ is compact then there is a contradiction. Ok, I know that every ...
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Prob. 6, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a normal space under a closed continuous map is also normal

Here is Prob. 6, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \to Y$ be a closed continuous surjective map. Show that if $X$ is normal, then so is $Y$. [Hint: If $...
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Prob. 3, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Every order topology is regular

Here is Prob. 3, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Show that every order topology is regular. First of all, here are some relevant definitions. Ordered Set: Let $...
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Prob. 2, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Any pair of disjoint closed sets in a normal space have neighborhoods whose closures are disjoint

Here is Prob. 2, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint. Here ...
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Prob. 1, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Every pair of points in a regular space have neighborhoods with disjoint closures

Here is Prob. 1, Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint. Here is ...
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If $X$ Hausdorff, compact with $G_{\delta }$ diagonal then $X$ has a countable basis

Suppose that $X$ is a Hausorff and compact space. Moreover, suppose that $\Delta:=\{(x,x):x\in X\}$ is a $G_{\delta}$ set. I need to show that $X$ has a countable basis. As $X$ is Hausdorff, $\Delta$ ...
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$X$ is regular iff $Y$ is regular

Let $q:X\to Y$ be a quotient function (i,e $q$ is continuous, onto and $U\subseteq Y$ is open iff $q^{-1}(U)$ is open). I need to show that $X$ is a regular topological space iff $Y$ is regular too. ...
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Separatedness of open subscheme of affine schemes

A scheme $X$ is separated if the daiganoal morphism $\Delta:X \rightarrow X \times_{\Bbb Z} X$ is a closed immersion. I know how to show that all affine schemes are separated. So Are open ...
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Example 2, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Normality of $\mathbb{R}_l$ — Why are these two sets disjoint?

The set $\mathbb{R}$ of real numbers with the lower limit topology having as a basis the collection of all closed-open intervals $[a, b)$, where $a, b \in \mathbb{R}$ with $a < b$, is denoted by $\...
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In a Topological Vector Space T0 implies T3½ (completely regular)? And other separation properties.

I will describe my doubt. I know that in a TVS T1 implies T2. Now since a TVS admits a uniformisable topology, we have that T2 implies the uniform structure is separating. Now a separating uniform ...
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Show by a counter example that a $(i,j)\gamma$-$T_1$ bitopological space may not be a $(i,j)\gamma$-$T_2$ bitopological space.

A bitopological space $(X, \tau_i, \tau_j)$ is said to be a $(i, j)\gamma$ -$T_1$ space iff for each pair of distinct points $x$ and $y$ in $X$, there exists $ij$-$\gamma$-open sets $G$ and $H$ ...
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Embedding of topological spaces into polytopes: complete regularity and metrizability.

In Dugundji's book Topology an interesting way to study topological spaces shows up frequerently: embedding them into polytopes which are defined by the author as arbitrary cartesian products of unit ...
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Separability and the Nagata-Smirnov Metrisation Theorem

Definitions: Let $X$ denote a topological space throughout. If all singleton subsets of $X$ are closed, then we call $X$ Fréchet. If, given any closed subset $C \subset X$ and any point $x \in X - C$...
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Behavior of points and compact subsets of Hausdorff spaces

It is quite straightforwad to see that many prpoperties are shared by points and compact subspaces of Hausdorff topologies, for example in terms of separation properties. I was wondering if there is ...
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Separating closed sets in Moore plane / Niemytzki plane (Topology)

I spent the last few days trying to solve this exercise with little success, so I really hope someone here might be able to assist: Denote Moore plane by $M$, the $x$-axis by $R$ and the upper ...
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Show that there exists a set $U$ which is both open and closed and $x \in U \subseteq V$.

Let $X$ be a compact topological space. Suppose that for any $x, y \in X$ with $x \neq y$, there exist open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively, such that $$ U_x \cup U_y = X\...