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

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2answers
118 views

Smooth, irreducible surface with real part containing two projective planes

Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...
2
votes
1answer
86 views

Measurability of integrals with respect to different measures

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
2
votes
0answers
303 views

Do Peano curves provide a counterargument to Grothendieck's critique?

This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...
2
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0answers
64 views

Separation-free topological completeness notion

Cannot really claim that I have immediate urgent motivation to study this question but it appeared to me long ago, I recalled it now by some reason and decided to ask it here. There is a strong ...
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1answer
97 views

Is the set of Cauchy spaces a lattice? [closed]

Is the set of all Cauchy spaces (ordered by set-theoretic inclusion) on some (fixed) set: a join-semilattice? a meet-semilattice? a complete lattice?
5
votes
1answer
275 views

Cardinality of connected Hausdorff topologies

Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then $(X,\tau)$...
7
votes
1answer
259 views

New separation axiom?

I am looking for the name and notation of the following separation axiom , temporarily denoted by $T_i$ (where $i=\sqrt{-1}$ is the imaginary unit): Axiom $T_i$: For any point $x$ of a topological ...
5
votes
1answer
121 views

Can There be Rudin-Keisler Immediate Sucessors?

There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...
1
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1answer
66 views

Partition of Real Number [closed]

Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
10
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1answer
606 views

Foundations of topology

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. Also some time ago I read ...
41
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1answer
1k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
4
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0answers
126 views

LCH topologies on Groups that are not group topologies

Ellis's 1957 paper on Locally Compact Transformation groups proves the following: A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...
10
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1answer
187 views

Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$). Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open ...
9
votes
2answers
228 views

Limits of rearranged sequences along ultrafilters

Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \...
7
votes
0answers
107 views

Are infinite simplicial complexes all manifolds?

Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic? Of course not. ...
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1answer
52 views

When is the orbit space of a manifold still a manifold of the same dimension?

$\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ ...
1
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0answers
59 views

Topologies with the same convex closed sets

Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
15
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2answers
639 views

$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?

An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter. Let $\mathfrak{ufo}$ be the minimal cardinality of an ultrafilter ...
14
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0answers
368 views

What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
1
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0answers
57 views

Properties of convergence at points of continuity

Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps. Then $J$ is a ...
1
vote
1answer
89 views

Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group $G$ by the left action of a maximal lattice 􀀀, i.e. a discrete cocompact subgroup....
2
votes
1answer
58 views

The pseudo-metric and linear orders

Is there a necessary and sufficient condition for a linearly ordered topological space to be pseudo-metrizable? (by a pseudo-metric, I mean a map $X\times X\rightarrow\mathbb{R}$ in which all the ...
0
votes
0answers
52 views

A question about Raikov complete topological groups

I asked this yesterday on Math Stackexchange, but didn't get any comments. So I thought I might ask here too. Let $G$ be a topological group and let $G^*$ be its Raikov completion, i.e its completion ...
4
votes
1answer
141 views

Stronger form of connectedness than path-connectedness

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(...
5
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1answer
246 views

References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
6
votes
3answers
394 views

When does the generalized Cantor space embed in a $\kappa$-compact space

The generalized Cantor space is the space $2^\kappa$, with basic open sets $$ [\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\}, $$ for $\sigma\in 2^{<\kappa}$. A space is $\kappa$-compact if ...
30
votes
5answers
2k views

Does $\mathbb C\mathbb P^\infty$ have a group structure?

Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$? $\mathbb C\...
21
votes
1answer
316 views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
2
votes
1answer
157 views

Pseudocomplements in the lattice of topologies

Given a set $X\neq \emptyset$ it is well-known that the collection $\text{Top}(X)$ of all topologies on $X$ is a (complete) lattice with respect to $\subseteq$. Let $0$ denote the smallest element of ...
2
votes
0answers
85 views

Sheaf on a filtered topological space?

Is there any nice way of defining a sheaf of abelian groups on a filtered topological space? Let $X$ equipped with filtration $X_0\subset X_1\subset X_2\subset ... \subset X_k=X$ be an object in the ...
3
votes
0answers
159 views

Every convex sequentially closed set is closed

Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed. Is there some description ...
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0answers
82 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
4
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1answer
184 views

Sequentiality of largest vector topology

I know that the largest vector topology on countable dimensional vector space is sequential (i.e. every sequentially closed set is closed). Does it keep for the arbitrary vector space? In countable ...
12
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3answers
260 views

$T_2$ topologies that are “as disjoint as possible”

Let $X$ be an infinite set. Are there Hausdorff topologies $\tau_1, \tau_2$ on $X$ such that $\tau_1\cap\tau_2 = \{\emptyset\} \cup \{U\subseteq X: X\setminus U\text{ is finite}\}$? (That is, the ...
7
votes
2answers
212 views

Reconstructibility of topological spaces

Let $(X,\tau), (Y,\sigma)$ be topological spaces with $|X|$ infinite and suppose $\varphi:X\to Y$ is a bijection such that for all $x\in X$ we have that $(X\setminus\{x\}) \cong (Y\setminus\{\varphi(x)...
0
votes
1answer
46 views

On 1-iso maps and subsets of the unit circle

Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
0
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1answer
92 views

Topology on $\omega\times\omega$ such that topologically connected equals graph-connected

For any set $X$ we define $[X]^2 =\big\{\{a,b\}: a, b \in X\text{ and }a\neq b\big\}$. Let $$E = \big\{\{(a_1, a_2), (b_1, b_2)\}\in[\omega\times\omega]^2: |a_i-b_i| = 1\text{ for some } i\in\{1,2\}\...
2
votes
0answers
45 views

Existence of enough local sections

Let $\pi: G\to X$ be a continuous open (!) surjection of locally compact Hausdorff spaces. Assume that each fiber $G_x=\pi^{-1}(x)$, $x\in X$ carries a group structure making it a locally compact ...
1
vote
1answer
98 views

Intersections of families of open sets ordered by well-inside relation in Euclidean space

Let $\langle\mathbf{R}^n,\mathscr{O}\rangle$ be the $n$-dimensional Euclidean space. Define $\mathbf{Q}\subseteq\mathcal{P}(\mathscr{O})$ to consist of all sets $\mathsf{Q}$ which simultanously ...
4
votes
1answer
122 views

$C(X)$ as finitely generated $C^*$-algebra

Let $X$ be a Hausdorff space. Suppose that $C(X)$ (or $C_0(X)$) is a finitely generated $C^*$-algebra. What we can say about $X$ ? For example can we characterize its inductive dimension, axioms of ...
6
votes
1answer
291 views

Classify $K(\pi,n)$ that are manifolds

Inspired by `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$. and When is a classifying space a topological manifold?, I'd like to formulate a precise question: For which $n \in \mathbb{Z}...
2
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0answers
61 views

Pre-cosheaf of connected components

Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...
4
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1answer
155 views

Is the topological dual of a Banach space weakly* closed in its algebraic dual?

The question is completely contained in the title :) I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
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3answers
537 views

What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
3
votes
1answer
109 views

If $S$ is non-stationary in $[k]^{\omega}$ is there a choice-function on $S$ with bounded fibers?

Fodor's Lemma : When $k$ is a regular uncountable cardinal, and $T$ is a stationary subset of $k$, any regressive $f:T\to k$ has a fiber which is stationary in $k$. Corollary: $T$ is stationary in $...
3
votes
1answer
209 views

When is the generalized Cantor space $\kappa$-compact?

My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references. The generalized Cantor space is the space $2^\kappa$, with basic open ...
4
votes
1answer
119 views

Is the following product-like space a Polish space?

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...
12
votes
2answers
257 views

Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity. For instance, a subset of ...
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0answers
167 views

A functor on the category of rings, algebras or compact Hausdorff topological space

Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra. We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
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0answers
100 views

Alternative representation of $C_c(X)$ as inductive limit

CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear. Under some additional constraints on the space (e.g. $X$ ...