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

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2
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0answers
19 views

Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...
3
votes
1answer
221 views

Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.
0
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0answers
40 views

First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions. We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
-2
votes
0answers
58 views

Complexity of a function [closed]

I am looking for a natural definition of the complexity a function. If the image is discrete, i was thinking it could be: consider the preimage of an element of the image, count the number of ...
13
votes
0answers
270 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on ...
2
votes
2answers
146 views

If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post). Basically following some ideas of W. Lawvere (but not his ...
4
votes
1answer
84 views

When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube

Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus ...
2
votes
3answers
180 views

Diffeomorphism with prescribed behaviour

If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...
4
votes
0answers
154 views

Why does $\beta \mathbb{R} \setminus \mathbb{R}$ have exactly 2 connected components? [closed]

Whilst reading about extensions of C*-alegbras, this topological fact was stated. I understand why $\beta \mathbb{R} \setminus \mathbb{R}$ has at least 2 connected components (it surjects onto the two ...
5
votes
0answers
75 views

Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
0
votes
1answer
44 views

Finiteness of “novel variance” from a kernel on a compact space [on hold]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
1
vote
1answer
154 views

Final topology of surjective linear map on Banach space

Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$. Is $\tau_L$ equivalent ...
1
vote
0answers
56 views

the validity of a basic statement involving the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$ Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k ...
4
votes
2answers
130 views

Tightness of Measures, Riesz Representation for locally compact spaces

Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, ...
12
votes
2answers
508 views

Low dimensional topological manifolds

There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: ...
1
vote
1answer
115 views

constructible set and fibre product

Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in ...
6
votes
1answer
186 views

Topological groups defined by completely disconnected subgroups

Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
6
votes
0answers
56 views

The space of all compact metric spaces with Gromov-Hausdorff distance

Given two metric spaces $(X_1,d_1),(X_2,d_2)$ one can define $d_{GH}(X_1,X_2)$---the Gromov Hausdorff distance between them. It appears to be $0$ iff $X_1$ and $X_2$ are isometric. One can therefore ...
0
votes
0answers
94 views

Krein-Rutman version of Hahn-Banach

Consider an arbitrary set of normed Riesz spaces $(X_i,\Vert \cdot \Vert_i,\leq)$, $i\in I$ ($I$ can be compact). Can I apply the Krein-Rutman version ( see Schaefer; TVS; Corollary 2 of 5.4, ...
1
vote
2answers
171 views

Jacobian of an injective mapping

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. I want to prove two things that seem intuitively right: 1) $f$ is ...
0
votes
1answer
61 views

dual space of the quotient space of some locally convex topological space

I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
8
votes
1answer
253 views

Must a closed totally path-disconnected subset of the sphere have connected complement?

This question (which is more a curiosity than a research problem) originates from these two: ...
13
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2answers
758 views

Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
1
vote
0answers
58 views

Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question. I think, if the following were true, then I would solve my problem. Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
1
vote
0answers
84 views

Relationship between weak Lp and strong Lq topologies for q<p

Specificaly: Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence? Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If ...
2
votes
1answer
105 views

A souped-up version of a question asked previously about uncountable subsets of topological spaces

Let T be an uncountable Hausdorff space. The following property of T will be referred to as "property P". If S is any uncountable subset of T, then the set of all points of S that are not limit points ...
16
votes
2answers
432 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
6
votes
3answers
341 views

Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
3
votes
1answer
184 views

Strange (?) definition of the spectrum

Suppose that $A$ is a commutative, unital $C^*$-algebra. Then it is isomorphic to $C(X)$ for some compact Hausdorff topological space $X$. $X$ can be identified as the space of all unital ...
0
votes
0answers
27 views

Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded?

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...
0
votes
0answers
32 views

Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
0
votes
1answer
87 views

Rational points in the Alexandroff line

Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line. Is $X$ a metrizable space?
2
votes
1answer
102 views

Connectedness properties of groups of homeomorphisms

Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...
0
votes
0answers
31 views

Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
3
votes
2answers
186 views

Non-Polish Lebesgue probability space?

Any Lebesgue probability space is mod. 0 isomorphic to some Polish probability space (with $\sigma$-algebra the completion of Borel algebra, and some Borel probability). I would like to see an example ...
6
votes
1answer
116 views

A question about cardinal functions

In the context of topological groups, one has $w(X)=d(X)\chi(X)$, where $X$ is a topological group and $w$, $d$ and $\chi$ are the weight, the density and the character, respectively. Since $d(X)\leq ...
1
vote
1answer
160 views

Tietze's extension theorem for compact subspaces

The topological question: Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set ...
6
votes
0answers
130 views

Zariski-homeomorphisms

This question is motivated by two questions at MO and at MSE. I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...
15
votes
1answer
449 views

Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
0
votes
1answer
214 views

Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
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votes
2answers
91 views

The boundary of this set is piecewise smooth? [closed]

Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean ...
2
votes
0answers
117 views

Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following Question. Let $X$ and $Y$ be metric spaces. When each ...
2
votes
0answers
71 views

The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability. As ...
0
votes
1answer
130 views

US does not imply AB

We say that a topological space $X$ is: $AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...
2
votes
0answers
45 views

A construction with Hyperspace of continums

Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...
2
votes
0answers
141 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
5
votes
2answers
206 views

Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure \begin{equation} H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...
0
votes
0answers
62 views

Problem on infinite dimensional metric space, with rigidity assumption

By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : ...
2
votes
1answer
109 views

A question about small cardinals related to Michael's Problem

I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times ...
0
votes
0answers
65 views

What do sparse sets in a norm topology look like in the weak* topology?

I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically, Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ...