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

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8
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10answers
2k views

Undergraduate Topology

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
2
votes
1answer
391 views
+100

Topological razors (ball-like spaces)

Introduction Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...
1
vote
0answers
68 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 $d: X \times X \to [0,\infty]$ ...
3
votes
0answers
44 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 ...
0
votes
1answer
288 views

Does there exist a countable partition of [0,1] by disjoint closed subsets? [duplicate]

Possible Duplicate: Why are the integers with the cofinite topology not path-connected? As in the title, is it possible to find closed, disjoint subsets $C_n$ of $[0,1]$ such that $[0,1] = ...
4
votes
1answer
289 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
3
votes
1answer
238 views

Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.
13
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0answers
299 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
159 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 ...
7
votes
2answers
386 views

Potential connected non-Lie subgroup

This painful question is inspired by the question "non-Lie subgroups" . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside ...
2
votes
3answers
185 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 ...
0
votes
1answer
385 views

Does anyone know an example of non-separable $L^1$ of a probability space?

It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy. ...
4
votes
1answer
85 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 ...
9
votes
0answers
240 views

Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

Is there a linearly Lindelöf space $X$ with $\operatorname{cf} (L(X))> \aleph_{0}$ (where $L$ is the cardinal function Lindelöf degree)? $L(X)$ must be a limit cardinal, like $\aleph_{\omega_{1}}$ ...
2
votes
0answers
172 views

Sequences and pseudocharacter in compact spaces

Is there a consistent example of a compact Hausdorff space $X$ on which the following holds? i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and ii) there is no non-trivial ...
0
votes
4answers
2k views

Does Cauchy continuity imply uniform continuity? [No.] [closed]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
1
vote
1answer
482 views

Example of a topological space

In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such ...
7
votes
4answers
1k views

Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff. What I am ...
1
vote
1answer
405 views

Topological properties of SpecMax(A)

We consider $A = C_{b}(X)$, the ring of continuous bounded functions on a completely regular space $X$. Let $\DeclareMathOperator{\SpecMax}{SpecMax} \SpecMax(A)$ be the set of maximal ideals of $A$ ...
2
votes
1answer
292 views

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint ...
5
votes
0answers
76 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 ...
9
votes
1answer
366 views

When do sheaves which are constant along the fibers come from the base?

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves ...
7
votes
1answer
533 views

A topological space for which having the ccc is independent of ZFC?

It is well known that a generalized Cantor space $2^A$ is separable if and only if $|A| \leq 2^{\aleph_0}$. This means that one cannot decide in $ZFC$ whether the space $2^{\omega_2}$ is separable or ...
4
votes
0answers
156 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 ...
33
votes
5answers
4k views

Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$ is used to represent the boundary operator in topology is that its behavior is akin to a derivative. But after ...
6
votes
2answers
337 views

How big $|Aut(M)|$ can be, given $|\partial Aut(M)|$?

My apologies: There were a couple of typos in the original question. Hope I got them all. Let $\kappa$ be an uncountable cardinal of cofinality $\omega$ and $M$ a model of size $\kappa$. We equip ...
18
votes
8answers
2k views

Connections between ultrafilters in topology and logic

I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ...
4
votes
1answer
344 views

“monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations. Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...
-1
votes
1answer
210 views

Topological degree of homogeneous function of degree k [closed]

Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k? This is true if F is ...
1
vote
1answer
155 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 ...
0
votes
1answer
46 views

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

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
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
131 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, ...
1
vote
2answers
172 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 ...
12
votes
2answers
512 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: ...
6
votes
1answer
189 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 ...
1
vote
1answer
116 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 ...
10
votes
13answers
4k views

Examples of non-metrizable spaces

I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other ...
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 ...
6
votes
0answers
57 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 ...
13
votes
4answers
994 views

Decidability of tiling R^2

Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane? I know the ...
0
votes
0answers
95 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, ...
19
votes
5answers
660 views

Topology on the set of analytic functions

Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$. Everyone who worked with this set knows that there is only one reasonable topology on it: the uniform convergence on ...
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 ...
3
votes
3answers
255 views

Complete uniform spaces require complete metrics?

Hey all, It is well-known that any uniformity is generated by the family of pseudometrics which are uniformly continuous from the product uniformity to $\mathbb{R}$. Further, the uniformity is ...
8
votes
1answer
254 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: ...
3
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0answers
214 views

How many ways do we have to prove that a mapping is open?

Given a continuous mapping $f$ between Euclidean domains (or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? ...
13
votes
2answers
762 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 $\widetilde D = \mathbb C \setminus D$ ...
2
votes
1answer
280 views

Equilibria Exist in Compact Convex Forward-Invariant Sets

Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : ...
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0answers
85 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 ...