**1**

vote

**2**answers

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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

**2**answers

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

**0**answers

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. ...

**-1**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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**

votes

**1**answer

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

**3**answers

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

**5**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**3**answers

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

**2**answers

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

**1**answer

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**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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 ...

**10**

votes

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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. $X$ ...