# Tagged Questions

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### Fundamental group of row of spheres [migrated]

The fundamental group of $S^1$ is $\mathbb Z$. Let's also call that space $P_1$. Then we'll build $P_n$ for $n > 1$ by taking $P_{n-1}$ and adjoining a circle to it with the condition that it must ...
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According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W ... 3answers 630 views ### When are maps between topological spaces homotopic? I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces$X$,$Y$(CW-complexes, say). So far I had the following ... 2answers 940 views ### Filters and intersection of two binary relations Let$\mathfrak{F}$is the complete lattice of filters (including the improper filter) on some set, ordered inverse to set-theoretic inclusion. I will denote$\left\langle f \right\rangle \mathcal{X} ...
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Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
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### Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site. Let $S$ be some orientable surface obtained by removing ...
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### On the cardinality of perfect spaces with the countable chain condition

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces? Recall that a ...
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### Topological characterisation for a (closed irreducible) hyperbolic 3-manifold

Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. ...
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### “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 ...
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### Defining a topology in the Power Set

I have the follwing question: Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$. If the ...
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### Fibration in the 3 torus.

The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between ...
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### Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$. Q: Does this imply that $U$ is homeomorphic to $U'$? In the case where the $\pi_1$'s are trivial then ...
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### Embedding Theorem for topological spaces, and in general

There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...
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### Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...
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I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies. DEFINITION   A continuous map   $u: ... 4answers 359 views ### Picturing a Certain Torus and Klein Bottle The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ... 1answer 424 views ### Showing a filter with a certain property on the power set of$\mathbb{Z}$is a one point filter Let$\mathcal{P}_0(X)$the Power set of$X$without the empty set and let$\dot{x}:=\{A\subseteq X: x \in A\}$the one point filter generated by$x$. Furtermore let $$\mathcal{A} := \{ f \in ... 1answer 278 views ### Contractibility of a configuration space For a topological space X and a positive integer k\in \mathbb{N}_{>0} let F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \} be its k-configuration space. Let ... 1answer 222 views ### Finding a good ordering of \mathbb{Q} Oftentimes in density arguments we let \{x_n\} be a dense sequence and this is sufficient to imply the desired result. From a research question I am working on I have simplified the ... 1answer 162 views ### Are period domains ever contractible Which simply-connected period domains are contractible? Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety? Are these contractible? 2answers 626 views ### Stone–Čech Compactification of \mathbb{Z} with Fürstenberg Topology The Stone–Čech Compactification of \mathbb{N} as a discrete space has been extensively studied and can be represented using ultrafilters. Consider X=(\mathbb{Z},\mathcal{T}), where \mathcal{T} ... 1answer 231 views ### Continuous functions on path-connected subsets Let X be a topological space, and PX the space of all paths on X. Then let G\subset X be a path-connected subset and p\in G a point. Let \sigma:G\rightarrow PX be a continuous function ... 3answers 585 views ### Convergence of probability measure and the *-weak convergence ? Given a Polish space X, I note C_b(X) the set of the continuous bounded functions with the norm of the uniform convergence, and and (C_b(X))^\star its topological dual with the *-weak ... 1answer 291 views ### sections of tensor product bundle ( tensor product of two vector bundles ) [closed] Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E ... 1answer 208 views ### Fluid mechanics and topology [closed] What are the intersections of fluid mechanics with topology especially the study of irrotational flow. A good answer should include references. 9answers 2k views ### Covering maps in real life that can be demonstrated to students Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ... 1answer 287 views ### Probability that a random distance function is metric Take a random n \times n nonnegative symmetric matrix D with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies D_{xy}+D_{yz} \geq D_{xz} for all index ... 2answers 237 views ### Homotopy Equivalences and Induced Correspondences between Fibre Bundles Suppose that f:X\rightarrow Y is a homotopy equivalence of manifolds. Given a manifold F, the pullback construction for f yields a correspondence between isomorphism classes of fibre bundles ... 2answers 305 views ### How to specify a finite group up to inner automorphism? I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ... 1answer 156 views ### A uniformity with a countable base is a pseudometric uniformity. I need a proof for this proposition: If a uniformity \mathfrak U on X has a countable fundamental system of entourages, then it can be defined by a pseudometric on X. which is the ... 1answer 189 views ### Group or manifold ? [closed] I have a question in seeing this$$U(n)=\frac{U(n)}{U(n-1)} * \frac{U(n-1)}{U(n-2)}*\cdots *\frac{U(2)}{U(1)}*U(1)$$So, group U(n) is written as product of quotient spaces. Is quotient space, for ... 1answer 191 views ### ( Homotopy) Y ENR and contractible subset => Y is a retract I'm trying to solve the following question: Y$\subset R^n$is a euclidian neighborhood retract. I want to prove that if$Y$is contractible it is a rectract of$R^n.$1answer 227 views ### Defining measures over frames in place of$\sigma$-algebras Normally measures and probability spaces are defined over$\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of$\sigma$-algebras? (i.e. complements do ... 1answer 241 views ### Homotopy groups of K3 Let X be a K3 surface and$Y=X/\mathbb{Z}_2$, an Enrique surface. Long exact sequence of homotopy groups corresponding to fiberaion$\pi:X\to Y$, says that$\pi_2(X)=\pi_2(Y)$, while we know$H_2(X)$... 1answer 431 views ### Example of a topological space… In my recent research, I defined a topological space$X$to be$EZ$-space if for every open subset$A$of$X$there exists a collection$\{A_{\alpha}: \alpha\in S\}$of clopen subsets of$X$such ... 1answer 580 views ### reference for “X compact <=> C_b(X) separable” (X metric space) I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ... 0answers 289 views ### Calculate the cohomology ring using Morse Theory [closed] My question is: Is there any way to calculate the cohomology ring of a finite dimension manifold using the Morse theory? The cohomology ring structure strongly rely on the following things ... 3answers 450 views ### Connection between properties of Dynamical and Ergodic Systems Hi All While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ... 2answers 295 views ### How big$|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$. We equip ... 3answers 367 views ### Topological proprieties of Specmax(A) Hello, We consider$A=C_{b}(X)$the ring of continous bounded fonction on a completely regular space$X$, SpecMax(A) the set of maximal ideal of$A$with the zariski topology. We know that there is ... 3answers 174 views ### Well-ordering with a topological property Assuming the axiom of choice, is there a well-ordering of the reals such that every initial segment is closed (for the usual topology)? If the continuum hypothesis helps we can also assume it. An ... 3answers 578 views ### Zero-dimensional space Let$X$be a topological space with the following property: for any open subset$A$of$X$there is a collection of clopen subsets$\{A_{\alpha}: \alpha\in S\}$such that ... 1answer 206 views ### What manifolds can have a (non-piecewise) linear structure? By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers$n$, I define$B_n$is to be$\: \big\{\mathbf{v} \in \mathbf{R}^n : ...
Let $X, Y$ be two metric spaces and $f$ be a continuous bijection (i.e. one-to-one map) from $X$ to $Y$. Let $E$ be a $G_{\delta}$ subset of $X$. I want to know weather the image $f(E)$ is also a ...