# 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 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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### 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 ...
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 $G$...