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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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$X$ 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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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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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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Is there anyone among us who can identify a certain SUS space?

The property US ("Unique Sequential limits") is a classic example of property implied by $T_2$ and implying $T_1$. In fact, it's the weakest assumption out of a chain of several distinct properties studied in the…
Steven Clontz
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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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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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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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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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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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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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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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Compact Hausdorff spaces are normal

I want to show that compact Hausdroff spaces are normal. To be honest, I have just learned the definition of normal, and it is a past exam question, so I want to learn how to prove this: I believe from reading the definition, being a normal space…
Hausdorff
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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…
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