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
2
votes
1
answer
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 ...
- 236k
-2
votes
0
answers
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 ...
- 622
0
votes
1
answer
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 ...
0
votes
1
answer
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 ...
- 2,931
1
vote
1
answer
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 ...
- 2,931
2
votes
1
answer
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 ...
- 3,143
5
votes
2
answers
125
views
How are k-Hausdorff and weakly Hausdorff distinct?
In this pull request to the pi-Base database, we encountered this situation.
A space $X$ is said to be weakly Hausdorff provided for every compact Hausdorff space $K$ and every continuous $f:K\to X$, $...
- 2,931
6
votes
1
answer
161
views
Why is a Countable Basis Needed in This Proof?
My question is regarding the Theorem in Munkres that states: Every Regular Space with a Countable Basis is Normal. Before reading the proof in Munkres, I tried to prove it myself and came up with a &...
- 177
1
vote
1
answer
53
views
Is there an error in Steen and Seebach's Counterexamples space 99 "Maximal Compact Topology"?
In Counterexamples space 99, the Maximal Compact Topology is asserted to be non-Hausdroff (note 1), KC (compacts are closed, note 3), and first-countable (note 5).
But KC spaces are US, and first ...
- 2,931
1
vote
3
answers
107
views
What separation is required to ensure extremally disconnected spaces are sequentially discrete?
Conversation continued at MathOverflow
In https://github.com/pi-base/data/issues/387 it is noted that Tychonoff extremally disconnected spaces are sequentially discrete (see Encyclopedia of General ...
- 2,931
5
votes
1
answer
37
views
How separated is a submetrizable space?
A space is submetrizable if its topology contains a coarser metrizable topology.
Each such space is Hausdorff: given two points $x,y$, the sets $B_{|x-y|/2}(x),B_{|x-y|/2}(y)$ are open in the space.
...
- 2,931
1
vote
1
answer
56
views
Are pseudonormal pseudocompacts all weakly countably compact?
All normal pseudocompacts are weakly countably compact (that is, infinite subsets cannot be closed and discrete).
Can "normal" be weakened to "pseudonormal"? A space is ...
- 2,931
2
votes
2
answers
108
views
Extremally disconnected without Hausdorff
In Theorem T000045 of pi-Base, a proof is given to defend the assertion from Counterexamples in Topology that all Extremally disconnected ($T_2$ where the closure of open is open) spaces are Totally ...
- 2,931
1
vote
0
answers
20
views
Definition of hereditarily collectionwise normal spaces using separated families
From some literature I've learned that there are two equivalent definitions of hereditarily collectionwise normal spaces, but struggling with finding a proof.
Definition 1. A topological space $X$ is ...
- 333
2
votes
0
answers
12
views
Continuous Functions on Compact T_2 topological spaces [duplicate]
Let $X$ be a compact $T_2$ topological space and let $x_0\in X$. Can we find a continuous function $f:X\to [0,1]$ such that $f(x_0)=1$ and $f(x) < 1$ for any $x_0\neq x$?
My try: Every Compact $T_2$...
- 195
1
vote
2
answers
69
views
Prove that a quotient space isn't Hausdorff
Let $Q=[0,1]\times[0,1]\subset \mathbb{R}^2$ and the following relation
$$(x,y)\sim (x',y')\iff (x,y)=(x',y'),\; (x',y')=(x,1-y),\;x\neq 0.$$
And $\pi: Q\to Q/\sim := X$ projection on the quotient. $X$...
- 728
4
votes
1
answer
88
views
Thomas plank is not realcompact
Let $X = \bigcup_{n\geq 0} L_n$ where $L_n = [0, 1)\times\{1/i\}$ for $i > 0$ and $L_0 = (0, 1)\times \{0\}$.
Define the topology on $X$ as follows: each point $(x, 1/i)$ for $x\in (0, 1)$ and $i &...
- 8,148
2
votes
1
answer
103
views
$T_D$ spaces and locally closed subsets
Let $X$ be a topological space, we say it is $T_D$ if each of its points is locally closed. That is, for any $x\in X$ we have $\lbrace x\rbrace=U\cap V$ where $U\subseteq X$ is an open subset and $V\...
- 2,005
1
vote
1
answer
30
views
A space is $T_3$ if every point has a closed neighborhood that is $T_3$ in the subspace topology
Let $(X, \tau)$ be a topological space. By $T_3$ I will mean that $\forall x \in X$ and $A $ closed subset of $X$ that does not contain $x$, there exists disjoint open sets $U, V$ such that $x \in U$,...
- 572
0
votes
0
answers
42
views
Proof Verification Half Open Intervals are Normal Topological Spaces [duplicate]
This is the question I am attempting: Show that the real line with the topology generated by the collection of half-open
intervals $[a, b) = \{x \mid a \leq x < b \}$ is normal topological space.
...
- 177
-1
votes
1
answer
38
views
How to construct counterexamples of spaces which satisfy some separation axioms?
I want to know some counterexamples of spaces which satisfy some separation axioms.For example, the counterexample for a space which doesn't satisfy $T_0$,$T_3$ and $T_4$ axioms.
Is there a more ...
-1
votes
1
answer
42
views
Locally compact and regular topogical space needs to be Hausdorff?
Consider X a topological space, that is locally compact and regular.
Is X necessarily a Hausdorff space?
Thank you in advance.
- 29
0
votes
0
answers
40
views
Regular Topological Infinite Spaces
Can anyone prove this please?
Let $Y$ be regular and $A\subset Y$ any infinite subset. Then there exists a family $\{U_n| n \geq 0\}$ of open sets whose closures are pairwise disjoint and such that $A\...
- 1
0
votes
0
answers
32
views
If {$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 ...
- 1
0
votes
0
answers
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 ...
- 281
3
votes
1
answer
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 ...
- 8,148
1
vote
1
answer
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 ...
- 2,931
0
votes
1
answer
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 ...
- 2,931
1
vote
1
answer
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 $x_n$.
...
- 2,931
2
votes
0
answers
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 $M$ ...
3
votes
0
answers
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} $ ...
- 12.7k
1
vote
1
answer
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 ...
- 3,713
3
votes
1
answer
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 $U$...
- 661
2
votes
0
answers
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 ...
- 1,697
0
votes
2
answers
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 ...
- 39
0
votes
2
answers
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 ...
- 10.9k
0
votes
1
answer
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 $X_{\alpha}\neq\emptyset$
3
votes
1
answer
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 ...
- 8,148
1
vote
1
answer
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 ...
- 4,666
0
votes
0
answers
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 ...
- 330
0
votes
0
answers
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 ...
- 4,666
1
vote
0
answers
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 $T_2$?
- 748
0
votes
0
answers
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 ...
- 728
0
votes
1
answer
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 ...
user733335
2
votes
1
answer
68
views
Prove that $(X,\tau_1\cap\tau_2)$ is also a $T_1$-space, whenever $\tau_1$ and $\tau_2$ are $T_1$.
Let $\tau_1$ and $\tau_2$ be two topologies on a set $X$ and that $(X,\tau_1)$ and $(X,\tau_2)$ are $T_1$-spaces (discrete topology). Prove that $(X,\tau_3)$ is also a $T_1$-space, where $\tau_3=\...
0
votes
0
answers
32
views
Separation axioms from a product to a particular space
I developed a proof of the following statement but I'm not sure if it is correct:
Proposition: Let $\lbrace X_\alpha \rbrace_{\alpha \in I}$ a collection of topological spaces and consider $X := \...
2
votes
2
answers
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 ...
- 10.9k
1
vote
0
answers
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) ...
- 309
4
votes
1
answer
86
views
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 ...
- 505
3
votes
0
answers
45
views
quotient space of similar matrices is not Hausdorff?
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 ...
- 61