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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.

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".

The most important separation axioms are $T_0$-spaces (Kolmogorov), $T_1$-spaces (Fréchet), $T_2$-spaces (Hausdorff), $T_{2\frac12}$-spaces (Urysohn), $T_3$-spaces (regular), $T_{3\frac12}$-spaces (completely regular) (Tychonoff) and $T_4$-spaces (normal) spaces.

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Does this property characterize a space as Hausdorff?

As a result of this question, I've been thinking about the following condition on a topological space $Y$: For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$…
Arturo Magidin
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$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed

Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. First, I tried to show that $X \times X \setminus D$…
Br09
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How to prove that a compact set in a Hausdorff topological space is closed?

How to prove that a compact set $K$ in a Hausdorff topological space $\mathbb{X}$ is closed? I seek a proof that is as self contained as possible. Thank you.
Elias Costa
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$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$

$X$ is a Hausdorff space and $\sim$ is an equivalence relation. If the quotient map is open, then $X/{\sim}$ is a Hausdorff space if and only if $\sim$ is a closed subset of the product space $X \times X$. Necessity is obvious, but I don't know how…
yaoxiao
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If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?

Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$. In a recent comment, I wondered whether $X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is closed for every continuous $f:X\to X$ (the forwards implication is a simple,…
Zev Chonoles
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When is $C_0(X)$ separable?

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I'm looking for a similar criterion for locally compact Hausdorff spaces, using $C_0(X)$ (the Banach space…
Paul Siegel
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How many compact Hausdorff spaces are there of a given cardinality?

This is a question I found myself wondering about recently. I eventually figured out the answer myself, but as this doesn't seem to be written down anywhere easy to find on the Internet I decided to share it here. Let $\kappa$ be an uncountable…
Eric Wofsey
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How to show that topological groups are automatically Hausdorff?

On page 146, James Munkres' textbook Topology(2ed), Show that $G$ (a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and $V \cdot y$ are disjoint. Noticeably, the…
Metta World Peace
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The product of Hausdorff spaces is Hausdorff

I'm confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, the basis elements consists of of products…
TopologyQuestion
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Is the closure of a Hausdorff space, Hausdorff?

$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
user59671
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Quotient Space of Hausdorff space

Is it true that quotient space of a Hausdorff space is necessarily Hausdorff? In the book "Algebraic Curves and Riemann Surfaces", by Miranda, the author writes: "$\mathbb{}P^2$ can be viewed as the quotient space of $\mathbb{C}^3-\{0\}$ by the…
user8186
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Examples for subspace of a normal space which is not normal

Are there any simple examples of subspaces of a normal space which are not normal? I know closed subspace of a normal space is normal, but open subspace in most cases which I can think of are also normal.
Jesse P Francis
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Do Hausdorff spaces that aren't completely regular appear in practice?

Completely regular spaces include all metrizable spaces, topological vector spaces, and topological groups in general. In fact, they are exactly the uniformizable spaces. Complete regularity is hereditary, ie. a subspace of a completely regular…
user54748
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Why are ordered spaces normal? [collecting proofs]

Greets This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will offer a Bounty in two days, depending on the…
Camilo Arosemena-Serrato
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Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there exists a unique continuous function…
Amr
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