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

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8
votes
2answers
274 views

Maximal trivialising subspace for a vector bundle

Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...
2
votes
0answers
199 views

algebraic structure of Integral Steenrod squares

It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations $$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$ In the case where $a$ is odd, one can define an ...
2
votes
1answer
109 views

Is the interval topology of $(\mathbb{N}^\mathbb{N}, \leq^*)$ connected?

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by $$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$ where $\downarrow x = \{y\in Q: y\leq ...
-1
votes
1answer
67 views

Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$

Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by $$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$ where $\downarrow x = \{y\in Q: y\leq ...
7
votes
1answer
294 views

A property stronger than the fixed point property

Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous ...
1
vote
1answer
97 views

A weak fixed point property

The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map. This motivates us to consider the following "weak ...
1
vote
0answers
90 views

Is there another equivalence relation on based maps between spheres which form the same graded ring as the homotopy groups?

Let $\sim$ be an equivalence relation on continuous based maps from $S^k$ to $S^n$, where $k$ and $n$ range over the positive integers. Suppose that Given maps $f, f^\prime: S^k \to S^n$ and $g, ...
5
votes
0answers
59 views

Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem. Let $G$ be a locally compact amenable ...
2
votes
1answer
182 views

Topologies for which the ensemble of probability measures is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess. ...
7
votes
0answers
131 views

Topology of family of complex varieties

It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that: For a proper flat map $f \colon X \rightarrow \Delta$, where $X$ is a complex ...
3
votes
1answer
130 views

Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point?

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the ...
2
votes
1answer
164 views

Is the set of entire functions Borel in the space of analytic functions?

$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm} $More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...
12
votes
3answers
289 views

Is a certain subset of the disc a convex set?

Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that? Draw a Cantor set $C$ on the circle ...
8
votes
0answers
139 views

Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff ...
1
vote
0answers
225 views

The closure of a set of closed points

Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...
1
vote
1answer
73 views

The separated uniform space associated with $(X,\mathfrak{U})$

If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in ...
2
votes
1answer
116 views

Is the complement of the ends of a manifold bounded?

Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...
2
votes
1answer
175 views

Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...
2
votes
3answers
260 views

Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let ...
6
votes
1answer
329 views

all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
5
votes
1answer
258 views

Which Banach spaces are realcompact?

I have a question about the topological space underlying a Banach space. A topological space $X$ is realcompact iff it is homeomorphic to a closed subset of an infinite product of the form $\mathbb ...
0
votes
1answer
78 views

Complement of a finite union of convex sets

Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components. I ...
1
vote
2answers
168 views

How do I prove that compact-open topology is metrizable?

Let $X$ be a $\sigma$-compact topological space and $(Y,d)$ be a metric space. Let $\{K_n\}$ be a sequence of compact subsets of $X$ whose union is $X$. Define $\rho_n(f,g):=\sup \{d(f(z),g(z)): ...
2
votes
2answers
194 views

Existence of a continuous section

Let $f\colon X\to Y$ be a surjective continuous map between two topological spaces such that $X,Y$ are path-connected and such that every fibre $f^{-1}(y)$ is connected, for each $y\in Y$. Is there ...
3
votes
1answer
90 views

Homotopy equivalence of maps with compact support and maps which vanish at infinity

Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space. What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence $$ \mathcal C_c(X,Y) \simeq ...
5
votes
0answers
80 views

Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, ...
6
votes
2answers
298 views

Is any function taking compact sets to compact sets, and connected sets to connected sets, necessarily continuous? [closed]

It is well-known that continuous image of any compact set is compact, and that continuous image of any connected set is connected. How far is the converse of the above statements true? More ...
12
votes
0answers
165 views

Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table. Let $$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$ for all $n\in\mathbb{N}$. Then since $C_{n}$ is a ...
6
votes
2answers
249 views

The role of the index set in the product of uncountably many topological spaces

Let $‎\langle‎ ‎‎X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology. Question. Is there a topological property that holds in ...
2
votes
1answer
201 views

Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]

This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points. DOUBT: Suppose $M\subset ...
8
votes
1answer
161 views

Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
6
votes
2answers
74 views

Closed field lines in the plane

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the ...
1
vote
1answer
186 views

Extension of continuous and smooth functions

Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of ...
6
votes
1answer
98 views

Countable subcover of half-open cylinders

While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book). Here I simplify the ...
41
votes
3answers
1k views

Duality between Compactness and Hausdorffness

Consider a non-empty set $X$ and its complete lattice of topologies (see also this thread). The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
3
votes
0answers
79 views

When closed subsets have finitely many connected componenets

Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?
9
votes
1answer
415 views

Is Max (R) a Hausdorff space?

I asked the following question at < http://math.stackexchange.com/questions/1508168/is-max-r-a-hausdorff-space >, but I pose it here for any help. Recall a space is totally disconnected if the ...
2
votes
0answers
110 views

Notion of convergence on a dense subset

My motivation for this question is as follows. Consider the set $D$ of cadlag functions on $(0,1)$ and its subset $D^\uparrow$ of non-decreasing cadlag functions. Each $f \in D$ has at most countably ...
5
votes
2answers
163 views

Existence of non-homeomorphic pair of bijectively related closed subsets in $\mathbb{R}$

I want to find two closed, non-homeomorphic subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections $$f:A\to B,~~~~g:B\to A.$$ Clearly ...
11
votes
1answer
243 views

A generalization of residual finiteness to topological groups

Consider the following generalization of residual finiteness to topological groups. A locally compact Hausdorff group $G$ is called residually compact if for every compact $K \subseteq G$ there is a ...
2
votes
0answers
109 views

Divisible fundamental group [duplicate]

I apologize if this question seems trivial or elementary. Is there any concrete topological space with divisible fundamental group? For example, is there any such a space the fundamental group in ...
4
votes
3answers
195 views

A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural ...
5
votes
1answer
85 views

Hausdorff open image of a Polish space

Let $f\colon X\to Y$ a continuous open and surjective function, where $X$ is Polish. It is known that $Y$ is Polish if: $f$ is closed or $Y$ is metric. Suppose that we know that $Y$ is Hausdorff, ...
17
votes
0answers
417 views

The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
10
votes
2answers
418 views

In CGWH, is every cofibration an inclusion with closed image?

As the title suggests, in CGWH, is every cofibration an inclusion with closed image?
9
votes
2answers
255 views

Two questions about the “grasp” cardinal function

For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the grasp $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a ...
7
votes
0answers
75 views

Locales satisfying DC?

Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of ...
5
votes
3answers
112 views

A Mackey-Ahrens theorem for uniform spaces?

Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = ...
5
votes
1answer
186 views

Existence of a non-null-homotopic simple closed curve

Assume that $X$ is a path-wise connected Hausdorff space, and assume that its fundamental group is non-trivial. Does it always exist a simple curve in $X$ which is non-null-homotopic? Such curve does ...
19
votes
3answers
408 views

Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...