# Questions tagged [gn.general-topology]

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

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### Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?

The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...

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128 views

### Disjoint covering number of an ideal

Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...

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247 views

### What is the local structure of a fibration?

It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled.
Here I'd like ...

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431 views

### Intersection form of logarithmic transformations

Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows.
Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$
be an ...

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120 views

### Has the Roelcke completion of a topological group any reasonable algebraic structure?

It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...

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141 views

### For which topological spaces X does an exponential object Y^X exist for all “nice” topological spaces Y?

A topological space $X$ is exponentiable, meaning that that an exponential object $Y^X$ exists for every topological space $Y$, iff $X$ is core-compact.
However, in the only proof I know of that non-...

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122 views

### Approximating an open bounded set by compact set

Let $U$ be an open bounded subset of $\mathbb{R}^n$, and $K$ a compact subset of $U$. Does there always exist a compact subset $L$ of $U$ that contains $K$, and such that $L$ is a retract of $U$.
...

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139 views

### Preservation of Baumgartner's I-ultrafilters under various forcings

For $I\subset \mathcal{P}(2^\omega)$, an ultrafilter $U$ on $\omega$ is said to be an I-ultrafilter if for all $f:\omega \to 2^\omega$, there exists $A\in U$ such that $f''[A]\in I$ [Baumgartner]. In ...

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136 views

### Do $G_\delta$-measurable maps preserve dimension?

This question (in a bit different form) I leaned from Olena Karlova.
Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...

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287 views

### Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...

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213 views

### On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...

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128 views

### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...

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68 views

### Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...

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209 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 ...

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165 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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132 views

### Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space
$X$ not greater than $2^{\omega}$?

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155 views

### Inverse limit in shape theory

Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...

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303 views

### Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...

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158 views

### Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$

Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$.
Are there group topologies $\mathcal A$ ...

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248 views

### Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...

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85 views

### Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...

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156 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 ...

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354 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? Here,...

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302 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 ...

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91 views

### Is a closed set with orbit capacity zero automatically thin?

Let $G$ be a countably infinite amenable group. Let $\alpha: G\curvearrowright X$ be a continuous group action. (Mostly free and minimal, though!)
Definition 1: Let $A\subset X$ be closed and $U\...

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111 views

### Equivariant zero dimensional extension recovering a given measure

Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action $\...

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277 views

### Fundamental group of non-Hausdorff surfaces & actions of discrete Heisenberg group

Let $G$ be a discrete group, acting on a space $X$ (by homeomorphisms). I will say that the action is properly discontinuous if for any $x, y \in X$, there are neighborhoods $U_x$ and $U_y$ such that ...

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124 views

### Possible homogeneity of infinite dimensional Sierpinski carpet analogues?

Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion.
Construct subsets $S_n$ by removing points from $H$ if for any $m$,
at least $n$ of the ...

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256 views

### Coloring $\mathbb{Z}^k$ and a fixed point theorem

This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...

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226 views

### Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...

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191 views

### Shrinking Group Actions

This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...

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481 views

### Profinite topologies

We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$.
My questions: Under ...

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191 views

### Is there a Whitney-type theorem Cauchy manifolds?

Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$.
Does it follow that there exists a subspace $N$ of $\...

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319 views

### Defining a topology by means of closed subsets in a topos

In the following we fix a topos. I'll speak of sets instead of objects and of subsets instead of subobjects.
Let $X$ be a set and assume $F$ is a set of subsets of $X$ that contains $\emptyset, X$, ...

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534 views

### continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...

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64 views

### Is the Baireness a 3-space property of topological groups

It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...

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172 views

### homotopy type of box topology.

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?

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104 views

### Maximal connected subtopologies

This is related to an older question.
Let $(X,\tau)$ be a topological space. Trivially, the indiscrete topology $\{\emptyset, X\}$ is a connected subtopology of $\tau$.
Is there a connected topology ...

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### The normality of powers versus the normality hypersymmetric powers

Let $X$ be a topological space. Let $[X]^{<\omega}$ be the space of non-empty finite subsets of $X$, endowed with the Vietoris topology. For a natural number $n$ the subspace $$[X]^{\le n}:=\{A\in[...

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370 views

### Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...

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### Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

I am not sure whether this question fits this forum (I will delete it if not appropriate here). But I asked this on MSE over a week ago with no answer and then put a bounty still got no answer. Here ...

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84 views

### point-wise approximation of the identity in hereditary Lindelof spaces

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...

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108 views

### A point concerning Fremlin's example on Borel sets in non-separable Banach spaces

Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...

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77 views

### A compact set in the plane with small sum-set and large projections

Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?
Remark. The ...

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188 views

### Set of subsequences with the same ultrafilter limit of the original sequence

Let $\mathscr{U}$ be a free ultrafilter on the positive integers $\mathbf{N}$ and fix $U \in \mathscr{U}$ such that $U$ is not cofinite (thanks J.D.Hamkins for the correction.)
Consider the natural ...

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184 views

### A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire,
if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...

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72 views

### Is $c(\mathcal M(X)) = c(X)$ for any first countable regular space?

G.M. Reed developed a construction technique which associates
a Moore space $\mathcal M(X)$ to each regular first-countable space $X$ such that $\mathcal M(X)$ is separable
(respectively, locally ...

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119 views

### Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber

Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...

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69 views

### When is the map $X^I \to X \times X$ open onto its image?

Here $X$ is a topological space, $I$ is the unit interval, $X^I$ is the function space and the map is evaluation at the end-points. According to the discussion at On the openness of the map X^I -> ...

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121 views

### Is an Abelian topological group compact if it is complete and Bohr-compact?

A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff.
A topological group $G$ is Bohr-compact if it admits ...