Mathematics

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.

968 questions
118
votes
4 answers

$X$ 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 $X \times X \setminus D$ is…
Br09
  • 2,320
111
votes
4 answers

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
  • 420,020
78
votes
5 answers

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
  • 15,400
59
votes
1 answer

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 literature: $T_2$ $k_1$ -Hausdorff…
Steven Clontz
  • 8,949
57
votes
6 answers

$X/{\sim}$ 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
  • 1,426
38
votes
1 answer

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
  • 9,607
36
votes
1 answer

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
  • 361
35
votes
3 answers

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
  • 133,510
35
votes
2 answers

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
  • 344,856
34
votes
3 answers

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
  • 6,723
  • 5
  • 38
  • 64
30
votes
8 answers

Examples of non-Hausdorff spaces

For instance, the simplest example of a non-Hausdorff topological space is the the pair $(X, \tau_X)$ where $\tau_X = \{ X, \varnothing \} $ . But this is boring. Can someone help me find more interesting examples?
ILoveMath
  • 11,129
28
votes
1 answer

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
28
votes
5 answers

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
28
votes
5 answers

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
  • 393
25
votes
6 answers

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
  • 1,571
1
2 3
…
64 65