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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Follow up question on union of disjoint Vitali sets...
Since I haven't received a satisfactory answer to my initial question I'm going to ask a somewhat weaker one...
This time we say $X$ is a Vitali set in the closed interval $[0, 1]$ with respect to $\...
- 54.2k
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Transversals to singular subvarieties
Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...
- 42.9k
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Weak convergence of measures on non-metrizable spaces
(ZF + Countable Choice)
Let $\langle X,\mathcal{T} \hspace{.06 in} \rangle$ be a second-countable Hausdorff space. Let $\mu$ be a Borel measure on $X$.
Let $\langle I,\leq_I \rangle$ be a directed ...
- 41.1k
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Weak homotopy equivalence of $H$-spaces
Suppose I have an $H$-space $H$ and a topological group $G$, such that for compact spaces $X$ there is a natural equivalence of group valued functors
$[X, H] \to [X, G]$
Now $H$ is a (non-finite) CW-...
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Shrinking Group Actions
This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...
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Representations of regular maps (four color theorem)
For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem.
For example, ...
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On Zariski Dense Subsets
Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of ...
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Connectifications?
Like many of my questions, this question is actually aimed at $p$-adic analysis.
One of the main obstacles in doing analysis $p$-adically ist that the $\mathbb{Q}_p$ is totally disconnected.
From ...
- 5,407
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2
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Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
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Topological spaces, uncountable subsets and separability
Hi, the following is a well known theorem
Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.
Question: Is there a similar result for topological ...
- 5,407
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Finite topological dimension x local compactness
Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...
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Can non-homeomorphic spaces have homeomorphic squares?
I an wondering if there are non-homeomorphic spaces $X$ and $Y$ such that $X^2$ is homeomorphic to $Y^2$.
- 52.3k
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Morse theory and Euler characteristics
Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...
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Induced pretopologies on sSet
Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
- 15.5k
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Is every closed set of Q² the intersection of some connected closed set of R² with Q²
Let $F\subset\mathbb{Q}^2$ a closed set. Does there exists some closed and connected set $G\subset\mathbb{R}^2$ such that $F=G\cap\mathbb{Q}^2$?
For example if $F=\{a,b\}$, you can take $G$ the ...
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Triviality of finite fiber bundles [closed]
Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...
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Some questions on Nicolai Reshetikhin's lectures on quantization of gauge theories.
This in reference to this fascinating lecture by Nicolai Reshetikhin-
http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf
Given what is said on page 13 in section 4.1 its not clear to me why ...
- 54.6k
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Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
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How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED
For referencing, I keep the original title and post and ask
only about the simplest case (and forget the freeway with crossings
for now).
Consider a trivalent graph, e.g. the dodecahedron or cube net....
- 56.6k
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space of homotopy equivalences of $S^1$
Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ ...
- 56.9k
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Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
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Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...
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What does the space induced by this unusual metric(?) on R/Z look like?
The motivation for this question comes from music theory. Dmitri
Tymoczko models "good" voice leading as minimizing distance between
pitches in successive chords. While this theory works well for ...
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Is there a classification of open subsets of euclidean space up to homeomorphism?
I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
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Faithful actions of finite groups on topological spaces
Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
- 15.5k
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Hausdorff-dimension of connected closed subsets of R^2
Let $A \subseteq \mathbb{R}^2$ be closed and connected, and for any $c \in \mathbb{R}$, let $A_c \subseteq A$ be a closed and connected subset of $A$, s.t. for $c \neq d$ we have $A_c \cap A_d = \...
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Universally measurable sets and weak topology
After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...
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Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups".
I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.
I am attempting ...
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Braid*Temperley-Lieb=?
I would be very astonished if this algebra isn't named.
You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and ...
- 28.1k
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Can a closed trefoil appear as a space-time "cut" of an open trefoil?
An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have ...
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Ring of continuous functions, reference request.
I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ ...
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On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
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Distinct, non-homeomorphic, profinite topologies on a given abstract group ?
Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...
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$\Delta_{2}^{1}$-hard set?
Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...
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Simple terminology question about the Dubrovnik (Kauffman) polynomial
In my S matrix classification attempts I encounter a lot of
Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n).
[Second variable is for writhe, n is an integer; for the first I don't
...
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Cantor set and Hilbert cube, or anything else?
I have recently rediscovered (after several years) the wonder of the Cantor set (so rich and so beautiful!). I have two questions that are unrelated, but they are both about Cantor sets.
Let $K$ be a ...
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Are there two non-homeomorphic finite regular CW complexes with isomorphic face posets?
Hello again! More of the same bumbling down the road of algebraic topology. This time, I am trying to figure out exactly how much information the face poset of a CW complex encodes. It has often ...
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Topological space associated to a real or complex scheme
Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...
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On Birman-Wenzlyfying the B2 spider
Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying
the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible
S matrices (paper pending) - it's ...
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Using topology to characterize embedded Lie subgroups of Lie groups.
Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup.
This leads us to ask the following question:
Can we replace "topologically closed" with a ...
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Lebesgue covering dimension
Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...
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Simple connectedness via closed curves or simple closed curves?
I've recently read some papers and books involving simply connected domains in Euclidean space (dimension at least 2), where domain is an open connected set. The usual definition is a (connected) set ...
- 25.1k
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compact-open topology
Is there a natural reason for defining the compact-open topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another ...
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Is there a "disjoint union" sigma algebra?
I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets $\{A_i\...
- 21.3k
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Numbers associated with boundaries of manifolds
I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...
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Is there a generalization of Brouwer's fixed point theorem?
In essence, this is the same problem as in
“The generalization of Brouwer's fixed point theorem?”.
But now I am determined to be careful. The main question is
the following:
Is there any ...
CommunityBot
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The generalization of Brouwer's fixed point theorem?
Let $X$ be a contractible compact [edit: locally connected] topological space
(Hausdorff and second countable). Let $f\colon X\to X$
be a continuous map. Then (I suppose) $f$ has a fixed
point. ...
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Minimal Hausdorff
A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff.
Every compact Hausdorff space ...
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Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
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Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
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