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

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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
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0answers
185 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
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1answer
354 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 ...
6
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3answers
657 views

Countable path-connected Hausdorff space

Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
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1answer
86 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
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1answer
57 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 $^{\mathbb S^n}/_{\mathcal D}$ ...
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2answers
396 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!
2
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1answer
70 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 $\prod_{i\in I} X_i$ ...
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0answers
85 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 ...
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69 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 ...
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1answer
52 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 $(X,\tau)\cong (Y,\sigma)$?
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140 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 ...
16
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2answers
960 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 $X\setminus \{x\}$ ...
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0answers
336 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 $Top$ ...
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127 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 ...
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95 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, ...
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48 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 $X_1$. ...
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0answers
99 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 ...
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1answer
87 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
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1answer
463 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 ...
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0answers
50 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 ...
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2answers
303 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 ...
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0answers
109 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 ...
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0answers
193 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
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2answers
180 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 ...
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0answers
56 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
167 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 ...
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0answers
140 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
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1answer
104 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 ...
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0answers
21 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 ...
9
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1answer
204 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 ...
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0answers
199 views

Existence of topology on the space of continuous functions

Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
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2answers
112 views

Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?

This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?
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2answers
211 views

Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?

If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
5
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1answer
138 views

(A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...
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3answers
1k views

Is there a subset of the plane that meets every line in two open intervals?

Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
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2answers
634 views

Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true: For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections. ( ...
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2answers
233 views

Concrete examples of covering from the 3-torus to the 3-sphere

There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...
5
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1answer
211 views

Does “$\forall Z(C(X,Z) \cong C(Y,Z))$” imply $X\cong Y$?

If $X, Y$ are topological spaces, let $C(X,Y)$ denote the collection of continuous maps $f: X\to Y$, endowed with the compact-open topology. Assume that we are given topological spaces $X,Y$ such ...
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0answers
74 views

Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point. Let us assume that $C_1$ and $C_2$ are two ...
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1answer
96 views

A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
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238 views

Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
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2answers
220 views

Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...
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2answers
376 views

Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
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4answers
327 views

When is the boundary of an open planar set a Jordan curve?

Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve? Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem. My ...
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2answers
115 views

Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$. Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths ...
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1answer
41 views

Compact $R_1$-spaces

A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$. If $X$ is compact and $R_1$, does ...
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0answers
34 views

Decomposition which is locally connected

It is possible construct a connected compact metric space $X$ and a continuous decomposition $\mathcal{G}$ of $X$ that satisfies: 1)$X/\mathcal{G}$ is locally connected. 2)If $M$ is a compact ...
3
votes
1answer
201 views

Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$. What is minimum ...
4
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2answers
214 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup ...