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 open convex PL subsets of R^n PL homeomorphic to R^n?
This is a basic issue of PL topology that I assume must be true, but I can't find a written reference: is a convex open PL subset of $\mathbb R^n$ PL homeomorphic to $\mathbb R^n$? I've scanned ...
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Concerning Problem 108 in "Open Problems in Topology"
In the book Open Problems in Topology by Jan Van Mill and George M. Reed, the following problem was presented:
108. Is there a para- Lindelof Dowker space?
Recall that a para-Lindelof Dowker space has ...
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Adjoining a new isolated point without changing the space
Suppose $X$ is a $T_1$ space with an infinite set of isolated points. Show that if $X^\sharp = X \cup \lbrace \infty \rbrace$ is obtained by adding a single new isolated point, then $X$ and $X^\sharp$ ...
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Continuity at a point in sequential spaces
Let $X$ be a sequential space, $Y$ be some topological space, and $f:X\mapsto Y$ define some function. If $\forall x_n \to x, n\in\mathbb{N}$ implies $f(x_n) \to f(x)$, then does it follow that $f$ is ...
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Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
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A Question about SO(n)
My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...
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What does the property that path-connectedness implies arc-connectedness imply?
A space X is path-connected if any two points are the endpoints of a path, that is, the image of a map [0,1] \to X. A space is arc-connected if any two points are the endpoints of a path, that, the ...
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Is every zero-dimensional space with no infinite clopen partition pseudocompact?
For this question, we say that a zero-dimensional space $X$ is $\omega$-pseudocompact if every partition of $X$ into clopen sets is finite. In other words, a zero-dimensional space $X$ is $\omega$-...
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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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Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
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Is every connected regular space having more than one point uncountable?
Is every connected regular space having more than one point uncountable?
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Weak compactness and weak sequential compactness in Banach spaces
If $E$ is a Banach space, $A$ is a subset of $E$ and is compact with the weak topology $\sigma(E,E')$, that is the most coarse topology which make every $f\in E'$ continuous, is it true that $A$ is ...
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Does local strict contractibility imply ANR?
Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...
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countably complete filters
Is there any description of the set of countably complete filters on the lattice of dense $G_{\delta}$ subsets of a compact, second countable metric space? [I haven't just dreamt this up: it describes ...
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Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
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What is a TMF in topology?
What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
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Continuity of the maxima
Let $f:X\times Y\mapsto R$ be jointly continuous, where $X$ is some topological space, $Y$ is some countably compact topological space. My question is whether or not the envelop $\phi(x) := \max_{y\in ...
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Uniform Embedding into Euclidean Space
Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset $Y\subset\...
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How (and when) to factor a function defined on a product of metric spaces?
Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor ...
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constructing a curve dividing two sets of points
Lets assume I have two sets of points, each characterized as being "A" or "B", respectively, that are in a Euclidean plane. Theoretically these two sets are samplings from a space that has ...
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Ring isomorphism $\Phi \colon C(X) \to C(Y)$ and zero dimensionality of $X$
We denote the ring of all continuous real-valued functions on $X$ by $C(X)$. The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $...
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Diffeomorphisms of a surface in terms of generators.
I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism (...
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What relates to measure spaces as topological spaces relate to metric spaces ?
Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
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In which cases a fiber bundle over a circle is a graph-manifold?
A fiber bundle over a circle $M^{3} \longrightarrow S^{1}$ with fiber a surface $F_{g}$ is characterized via a homeomorphism $\varphi \colon F_{g} \to F_{g}$. It can be one of the following: periodic, ...
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Existence of an arbitrary Small positive continuous real Valued Function
Let $(X,\tau)$ be a Tychonoff Topological space.
For each $x\in X$ consider an arbitrary positive real number $\epsilon_x>0$. Is There a continuous real valued function $f:X\rightarrow \mathbb{...
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Bitopological spaces and algebraic topology
Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy ...
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On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
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Homology and homotopy of a surface
Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...
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On Pseudo-finite topological spaces
We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an ...
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What information can one recover from the induced map on homology?
The following question came up while constructing delay embeddings of time series data.
Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a ...
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Products with compactly generated spaces
It is well known that if $X$ and $Y$ are topological spaces with $X$ locally compact Hausdorff and $Y$ compactly generated, then $X \times Y$ (with the ordinary product topology) is compactly ...
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About subspaces of $F$-spaces
A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" ...
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Example of a compact homogeneous metric space which is not a manifold
A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
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Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
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Ring structrures on R^n
Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non ...
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Level 2 Menger Sponge
Hi fellows,
Does anyone know the number of holes of a level 2 Menger Sponge ?
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When a set of measure zero plus itself contains interior
Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$.
Is it true ...
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On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.
Property A ...
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F-spaces and points whose complements are C*-embedded
Let $X$ be a compact Hausdorff space. If $X$ is extremally disconnected then the complement of every point $x$ in $X$ is C*-embedded in $X$ (i.e every continuous bounded real-valued function on $X\...
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Some Questions about zero-dimensional subsets of the unit interval related to cantor set
Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
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Does the "measure-preserving property" commute with ultralimits ?
Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and ...
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Existence of a non-submetrizable topological space $(X, \tau)$
We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.
one of the properties of these topological spaces is ...
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How do you call a map which sends convergent sequences to pre-compact ones ?
In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ ...
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Abstract definition of properly discontinuous action
A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract ...
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Ideals of $C(X)$ with only finitely many number of zerosets
We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where ...
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Homomorphisms of Topological Groups which are Automatically Fiber Bundles?
Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
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Category of Uniform spaces
I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and cartesian closed. Can anyone comfirm or deny, ...
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Counterexample about Jones lemma with special weak condition.
Jones Lemma is One scale about recognizing that a topological space is not normal.
This lemma tells us, if The topological space $X$ has a dense subset $D$ and a closed discrete subset $S$, with the ...
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Does this knot invariant distinguish trefoil chiralities?
Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently ...
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Locally compact, 0-dimensional, pseudocompact space
Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which ...
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