Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

learn more… | top users | synonyms (2)

1
vote
1answer
44 views

Does order-preserving equal continuous? [on hold]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
3
votes
0answers
28 views

Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$? (This is a follow-up ...
5
votes
1answer
68 views

Path-connected interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?
5
votes
1answer
122 views

Properties of the interval topology of the lattice of functions

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: ...
11
votes
1answer
174 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
3
votes
0answers
83 views

Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen: TVS's $V$ of ...
3
votes
1answer
97 views

Quotients of powers of the Sierpinski space

Is every space isomorphic to some quotient of a power of the Sierpinski space? More precisely: Let $(X,\tau)$ be a topological space, and let $\mathbb{S} = (\{0,1\}, \{\emptyset, \{0\},\{0,1\})$ be ...
-2
votes
0answers
38 views

interior, closure, continuity, composite in topology [closed]

Show by a counter example that the function f which assigns to each set is interior does not commute with the function g which assigns to each set is closure.
-5
votes
0answers
123 views

A beginner's next topology book: what direction after intro to topology text book? Also Measure Theory [closed]

I am about to finished Mendelson's Introduction to Topology and I would like to know what should be my next book? (I am working on my own and am not affiliated with any universities.) This one? A ...
2
votes
0answers
61 views

Quotients of simplicial complexes which are simplicial complexes

In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...
4
votes
1answer
101 views

Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense: A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...
1
vote
0answers
153 views

Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to ...
3
votes
0answers
41 views

Injectively rigid spaces

Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map? (This is Joel David Hamkins's recent question in the category ...
1
vote
0answers
170 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
0
votes
1answer
56 views

Corresponding between prime ideals in $C(X)$ and $C^*(X)$

we know that every maximal ideal in $C(X)$ is in this form: $$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$ and every maximal ideal in $C^*(X)$ is ...
-1
votes
0answers
66 views

diagonal stratum in symmetric product

Fix a Riemann surface $(\Sigma, j )$ and a partition $\pi: r=\Sigma a_in_i$ for integer $a_i \ge 1, n_i \ge 1$, there is a diagonal stratum $\chi_{\pi}$ indexed by $\pi$ comprising the image of the ...
0
votes
1answer
47 views

Rank of a generall linear group over a finite field [closed]

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.
15
votes
3answers
425 views

Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
4
votes
1answer
116 views

Reconstructing relations with the image relation of a topology

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive. This ...
3
votes
1answer
52 views

Hausdorff spaces with asymmetric image relation

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive, and ...
5
votes
0answers
122 views

Topological Subset Take-Away

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none ...
15
votes
1answer
313 views

The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
7
votes
3answers
546 views

Countable path-connected Hausdorff space

Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
2
votes
1answer
76 views

“Universal” connected spaces

Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true? ...
2
votes
1answer
46 views

Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...
10
votes
1answer
298 views

Connected but no path connected components

Is there a Borel subset of plane which is connected but whose only path connected components are singletons? I know that a Bernstein set is a non Borel example of such a set. Thanks!
3
votes
0answers
39 views

Is the covering property $\Omega \choose \text{T}$ closed under products?

Suppose that $(X_i)_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see this post). Does $\prod_{i\in I} X_i$ ...
1
vote
0answers
81 views

When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
6
votes
0answers
60 views

Hausdorff spaces with lattice isomorphism between the topologies [closed]

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic. Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$? (This is a follow-up question to ...
2
votes
1answer
43 views

$T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$. Does this imply that $(X,\tau)\cong (Y,\sigma)$?
8
votes
0answers
134 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here. The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...
18
votes
2answers
899 views

Non-homeomorphic spaces such that taking away a point makes them homeomorphic

Are there topological spaces $X,Y$, each having more than $2$ points, such that $X\not\cong Y$, and there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ the spaces $X\setminus \{x\}$ ...
8
votes
0answers
300 views

Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...
5
votes
0answers
108 views

Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a “Wick rotation”?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like ...
2
votes
0answers
87 views

cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
1
vote
0answers
44 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
5
votes
0answers
68 views

Surjectivity of self-isometries as property of metric spaces

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
1
vote
1answer
71 views

Bounded metric spaces with non-surjective self-isometry

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
13
votes
1answer
394 views

On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at ...
1
vote
0answers
49 views

Extending an homotopy, knowing the two base functions extend

Let $A\subset B$ be paracompact spaces, and let $C$ be a paracompact space. Let $f_0,f_1:A\rightarrow C$ be continuous functions. $F:A\times[0,1]\rightarrow C$ a homotopy from $f_0$ to $f_1$. Suppose ...
6
votes
2answers
228 views

Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
1
vote
0answers
82 views

How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...
2
votes
0answers
171 views

Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
5
votes
2answers
175 views

If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?

In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following. Let $\mathbb{IR}$ be the interval domain $\lbrace ...
2
votes
0answers
47 views

Are all locally compact anisotropic groupoids etale up to equivalence?

By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ...
5
votes
1answer
158 views

Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have. Is there a non-separable complex manifold? Can a non-separable complex ...
0
votes
0answers
118 views

Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be ...
2
votes
1answer
96 views

Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales? Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales. For two locales, $A$ and $B$, is there a nice way to make an ...
1
vote
0answers
11 views

Defining connectivity between K points on a periodic domain in terms of proximity

THE SITUATION: Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...
6
votes
1answer
170 views

What is $Sq^i(\alpha^j)$ for all $i$ and $j$?

Write $H^*(\mathbb{R}P^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[\alpha]$, $\deg \alpha = 1$. What is $Sq^i(\alpha^j)$ for all $i$ and $j$? I am not an algebraic topologist by trade but need to know this ...