Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,602 questions
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Is a measurable homomorphism on a Lie group smooth?
Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...
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An open problem on general topology
There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.
Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $...
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Measures idempotent with respect to addition and multiplication.
Does there exist a probability finitely additive measure on $\mathbb N$ which is idempotent with respect to addition and multiplication simultaneously?
It is known (due to Hindman) that there is no ...
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Example of a completely regular spaces
A topological space $X$ is an $EF$-space if if for any
two collections $\mathcal{U}$ and $\mathcal{V}$ of clopen subsets
of $X$ with $\bigcup \mathcal{U}\cap \bigcup
\mathcal{V}=\emptyset$, we have $\...
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Is there a good concept of a measurable fibration?
In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability ...
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realcompact space
I want to study realcompact spaces but I can't find the best book or journal for it, and I really need to.
(sorry I don't write English very well)
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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 $H_2(X)$ ...
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Is there a long exact sequence associated to a ramified covering?
A covering map $p:X\to Y$ between topological spaces can be viewed as a fiber bundle $\Sigma\to X\to Y$ with a discrete group $\Sigma=Gal(X/Y)$ as fiber. Such a fiber bundle leads to a long exact ...
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Noncontractible connected topological rings ?
Are there any non-contractible connected topological rings?
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by Lurie ...
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A question on hereditary Lindelof number
Let $X$ be space. A space $X$ is called right-separated if it can be well-ordered in such a way that every initial segment is open in $X$. See the related link (left-separated).
How could we show ...
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Understanding the left-separated spaces
A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.
Could someone post some left-separated space to help me understand such ...
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Is there a (standard) name for $\bar{A}\setminus A$?
This is a notation question:
If $A$ is a set in a topological space and $\bar{A}$ is its closure, is there a (standard) name for $\bar{A}\setminus A$?
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Connectedness of the complement of small subsets (extended question)
The following questions occurred to me while browsing this site and looking at Exercise 20 here.
Question 1. Let $n>1$. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which ...
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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 ...
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Forcing over the poset of nonempty open subsets of a nice topological space
Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
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Roadmap to Complex Dynamics (Particularly the works of Hubbard, Douady, and Yoccoz regarding the Mandelbrot set)
As others have had great success with their question, I hope to ask one in a similar vein. As a student who has some background in complex analysis and dynamical systems, I am hoping to explore ...
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continuous R^2xR^2xR^2/E^+(2) -> R^3 injection?
This is a question that comes from my (biological) research. I'm very weak in topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two ...
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Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
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Uniqueness and existence of topology for a given convergence class of nets
My question comes from reading Pete Clark's reply How do you axiomatize topology via nets?
In the section "Convergence Classes" at the end of Chapter 2 of his book, Kelley lists the ...
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Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?
A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...
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How to prove the isotopy relative to a point exist?
Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
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perfect space without convergent long sequences
Is there a boolean space $X$ without isolated points with the property that no point $x\in X$ is the limit of a long sequence $(x_i)_{i\in I}$ from $X\setminus \lbrace x\rbrace $ ('long sequence' here ...
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Automatic continuity of the inverse map
All topological spaces considered here are Hausdorff.
It is a well-known consequence of the minimality of a compact topology that an injective continuous map
$f\colon X\to Y$
where $X$ is compact, ...
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Is the wedge sum of two cones over the hawaiian earring contractible?
Let $X_1$ and $X_2$ be two cones over the hawaiian earring and let $X$ be the wedge sum of $X_1$ and $X_2$ (of course you join them in the special point of the hawaiian earring). How do you prove that ...
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Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
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Type I subspaces of the Stone Cech compactification of $\omega$
EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested.
A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
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Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
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How should one look at the set of compatible ring structures on a given group?
Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...
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Complexity of a fixed point
Let $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a homeomorphism of
the plane with fixed point $p$, i.e. $\varphi(p)=p$, and no other periodic
points. Let $r$ be a fixed natural number. My ...
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Does a uniform space have a closed embedding in a product of metric spaces?
I am assuming that uniform spaces are Hausdorff (although it probably doesn't matter for this question). It is more-or-less obvious that a uniform space can be embedded in a product of metric space (...
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Descriptive set theory on $\mathbb{R}^\mathbb{N}$
The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
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function space in comma category
Let TOP be a category of topological spaces and B be an object of TOP. Is there a notion of function space in the comma category TOP/B.
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Quotients of Cantor cubes onto spaces
Let $\lambda$ be an infinite cardinal. Consider the Cantor cube $\Delta_\lambda = \{0,1\}^\lambda$. It is a standard fact in topology that the topological weight (= minimal cardinality for a basis) of ...
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Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
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Counterpart of Weierstrass theorem
Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is ...
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Algebras with countable chains only
Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
- 3,937
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Quotient rings of $C(X)$
Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
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Are amenable groups topologizable?
I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
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When does a homeomorphism split into essentially minimal homeomorphisms?
Background
Suppose $X$ is a compact metric space, and that $\varphi: X\to X$ is a homeomorphism of $X$.
We say a subset $A$ of $X$ is $\varphi$-invariant if $\varphi(A) = A$. A $\varphi$-invariant ...
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special extremally disconnected spaces with only finite isolated points
We Know that a cardinal $\kappa$ is measurable if there is a set $X$ with cardinal $\kappa$ and a {0,1}-measure $\mu: P(X) \rightarrow ${$0,1$} so that for all $x \in X$, $\mu(x)=0$ and $\mu(X)=1$. ...
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Does this construction yield an injective hull ?
Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as ...
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Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
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Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
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Composition of (topologically) connected binary relations
My question seems far too basic to be unknown, but I could not find anything relevant...
Let $X$, $Y$ and $Z$ be compact connected metric spaces, and let $F \subset X \times Y$ and $G \subset Y \...
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can an nonzero IC sheaf have zero hypercohomology?
Can someone tell me which of the following are true? Let $X$ be a reasonable space.
Suppose $F$ is a complex whose cohomology groups are constructible sheaves, at least one of which is nontrivial.
...
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Applications of Brouwer's fixed point theorem
I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:
...
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Is the generalized Baire space complete?
I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy ...
- 2,754
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On the image of a G_\delta set under a continuous bijection
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 $G_{...
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A question about local connectedness
Let C be a connected and completely metrizable subset of the Euclidean plane. Can C fail
to be locally connected at each of its points?
- 6,548
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Compactification of topological spaces
Hello,
If we take a localy compact space $X$ and we put $A=C_{b}(X)$ the $\mathbb{R}$-algebra of bounded continous functions on $X$, we have an embeding of topological space
$$\psi:X\longrightarrow ...
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