# 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.

858 questions
Filter by
Sorted by
Tagged with
46 views

### Comparing two notions of "needed-for-Hausdorffness"

Suppose $\mathcal{X}=(X,\tau)$ is a non-Hausdorff space. There are two ways to make precise "the sets that need to become open if we refine $\mathcal{X}$ to become Hausdorff" that I'm ...
88 views

### Uncountable product of real interval is connected and separable? [closed]

I've learned that countable product of connected separable space is connected separable. Suppose we have an arbitrary set $S$. Let $A=[0,1]^S$ be a weakly ordered space. Is $A$ connected separable? A ...
33 views

### Proof that every $T_0$ Topological Vector Space is regular

I'm currently reading through Convex Analysis and Beyond by Mordukhovich and Nam where the following preposition and proof are given (note that the authors define TVSs to be $T_0$). Proposition 1.92 ...
31 views

### $k_3$-Hausdorff

A subset of a space is $k_1$-closed if its intersection with every compact subset is closed in the subset. A subset of a space is $k_2$-closed if its intersection with every continuous image of a ...
1 vote
33 views

### Are retracts always closed in a compact weak Hausdorff space?

A retract is a subspace $R\subseteq X$ for which there exists a continuous map $f:X\to R$ with $f(r)=r$ for all $r\in R$. If $X$ is compact and has the property KC (all compacts are closed, a ...
39 views

### Is a space $T_1$ iff every irreducible component is a point?

I've been thinking a little about irreducible components while reading Atiyah and Macdonald and wanted to try and understand the idea of irreducible component in more general terms of separation ...
125 views

103 views

32 views

### If {$x$}' is closed then {$x$}' is the union of disjoint closed sets.

I'm reading about separation axioms weaker than $T_1$ on topological spaces. One article makes the following claim: In an arbitrary topological space $(X,\tau)$ (Not necessarily $T_1$) "If for ...
39 views

### If $X$ is regular and $f:X \to Y$ is a continuous, surjective, open and closed function, then $Y$ is Hausdorff.

Let $(Y, \sigma)$ and $(X, \tau)$ be two topological spaces. If $X$ is regular and $f:X \to Y$ is a continuous, surjective, open and closed function, then $Y$ is Hausdorff. I've already proved that if ...
54 views

### Continuity using nets on a dense subspace and regularity

If $Y$ is regular, $f:T\to Y$ is a function such that $X\subseteq T$ is dense in $T$, and for any net $x_\alpha\in X$ converging to $x\in T$, $f(x_\alpha)$ converges to $f(x)$, then $f$ must be ...
1 vote
44 views

### Fully normal implies collectionwise normal?

T214 of pi-Base tracks that fully $T_4$ spaces are collectionwise normal, citing an assertion from a diagram in Steen/Seebach's Counterexamples. Is fully normal sufficient? In any case, how can this ...
35 views

### Perfect but not fully normal spaces?

A perfectly normal space satisfies that all closed sets are $G_\delta$ (countable intersections of open sets). This implies complete normality. A fully normal space satisfies that every open cover may ...
1 vote
49 views

### What separation properties are satisfied by the Arens space?

The Arens space is obtained by taking a discrete sequence $\{x_n:n<\omega\}$ converging to $\infty$, then attaching another discrete sequence $\{x_{n,m}:m<\omega\}$ converging to each $x_n$. ...
21 views

### Open "enlargement" of locally finite family of compact sets

I want to show that Let $M$ be a paracompact $T_2$ space and let $(K_i)_{i\in I}$ be a locally finite family of compact subsets of $M$, then there is a locally finite open covering $(U_j)_J$ of $M$ ...
191 views

### Spaces where each compact subset has compact closure: have they already been studied?

Let $(X, \tau)$ be a topological space be such that for every $K\subset X$ compact implies $\operatorname{cl}(K)$ is also compact. I am interested in the study of such spaces [call $\textrm{G}$ ...
1 vote
132 views

### How to show that an unbounded set does not exist using the axioms of ZF Set Theory?

I am currently studying a course on ZF Set Theory and would like to know how to show that a set does not exist using the axioms. For example, the following two sets are clearly equivalent to a set of ...
59 views

### Proof Verification: Compact Hausdorff Implies Normal

Here is the definition of "normal space" that I am working with: A topological space $(X,\tau)$ is normal if for any two disjoint closed sets $A$ and $B$, there exists disjoint open sets $U$...
52 views

### How many second-countable $T_1$ spaces are there? [duplicate]

How many second-countable $T_1$ spaces, up to homeomorphism, are there? Let $X$ be a second-countable $T_1$ space. Since $X$ is second-countable, there are at most $\beth_1$ open sets in $X$. And ...
68 views

### Is Right ray topology is normal space?

Let $\mathbb{R}$ be the set all real numbers with right ray topology $\Im$={$(a,\infty):a\in \mathbb{R}$}$\cup${$\mathbb{R},\emptyset$}. Is $(\mathbb{R},\Im)$ a normal space? Each non-empty closed set ...
167 views

### How to show that the space $Y$ obtained by identifying points on the same straight line through origin in $R^{n+1}-\{0\}$ is Hausdorff?

Background: $Y$ turns out to be projective space but I'm trying to prove this and have not proven it yet (i.e., $Y$ is homeomorphic to the space $X$ obtained by identifying antipodal points on the ...
47 views

### $X_{\alpha}$ is normal when $\prod{X_{\alpha}}$ is normal

If $\prod{X_{\alpha}}$ is normal in product topology then $X_{\alpha}$ is normal for all $\alpha$ assuming $X_{\alpha}\neq\emptyset$
83 views

### Separating countable sets and Urysohn spaces

Consider the following separation property: For any countable disjoint subsets $A, B\subseteq X$ of a $T_1$ space $X$ such that $A\cap \overline{B} = \overline{A}\cap B = \emptyset$ there exist ...
1 vote
83 views

### Prove that if $Y$ is hausdorff then $Y^X$ is it too when it is equipped with open-compact topology.

So in this question is asked to show that if $Y$ is an Hausdorff space then for any other space $X$ the subset of continuous function $\mathcal C(X,Y)$ of $Y^X$ is Hausdorff but I think that really ...
80 views

### Geometric realization of a simplicial set is Hausdorff

I am reading the book May's Simplicial objects in algebraic topology and I am trying to understand the proof for Theorem 14.1 - the geometric realization of a simplicial set is a CW-complex. He says ...
64 views

### Is a fully normal space even normal?

Ryzard Engelking in his topology text at the historical and bibliographic notes of $5.1$ says that a topological space $X$ is fully normal if every open cover has an open star refinement (click here ...
1 vote
22 views

### Example of closed map from $T_2$-space onto non-$T_2$ quotient? [duplicate]

What is an example of a continuous closed surjection $f : X \rightarrow Y$ from a $T_2$-space $X$ to a space $Y$ that is not $T_2$?
30 views

### What is this separation property of locales called

Trying to come up with a point-free formulation of the $T_1$ axiom I thought of the following condition on a frame/locale $L$: \forall U \in L \, . \, U \wedge \bigwedge \{ V \in L : U \vee V = \top ...
47 views

### A type of $T_2$ topological space such that for every point there's a $T_3$ subspace then the whole space is $T_3$ [closed]

Given a $T_2$ topological space $(X,\tau)$ such that $\forall x\in X, \exists V\in \tau$ such that $x \in V$ and the subspace $(cl(V),\tau_{cl(V)})$ is $T_3$ then $(X,\tau)$ is $T_3$. I really don't ... 68 views

100 views

### Why is a perfectly normal space completely normal?

Suppose that $X$ is perfectly normal space. To show that $X$ is completely normal, I must show that every subspace of $X$ is normal. To that effect, let $Y$ be a subspace of $X$. Let $A, B$ be ...
1 vote
32 views

### A problem about paracompact Hausdorff space and its closed subset.

Let X be a paracompact Hausdorff space. $M=\bigcup_{i=1}^{\infty}F_i$, $F_i$ is closed. Prove that M is paracompact as a subspace of X. Idea: In Munkres' book topology, there are two theorems: (1) ...
### If Any continuous map from $A$ into $\mathbf{R}$ may be extended to a continuous map of all of $X$ into $\mathbf{R}$. Then $A$ is closed subset.
TRUE/ FALSE: Let $X$ be a normal space and $A$ be a subspace of $X$. If Any continuous map from $A$ into $\mathbf{R}$ may be extended to a continuous map of all of $X$ into $\mathbf{R}$. Then $A$ is ...
This is a problem of my midterm test... Denote invertible $2\times 2$ matrices on $\mathbb{C}$ by ${\rm GL}(2,\mathbb{C})$, and define the conjugation action of ${\rm GL}(2,\mathbb{C})$ on itself by ...