All Questions
5,185 questions
1
vote
1
answer
121
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More general metric spaces (where image of metric is not a subset of $\mathbb R$)
When defining metric spaces we want that $d$ in a pair $(X,d)$ satisfies:
1) $d$ is a function from $X \times X$ to $\mathbb R$
2) $d(x,y) \geq 0$ with $d(x,y)=0$ iff $x=y$
3) $d(x,y)=d(y,x)$
4) $...
- 225
1
vote
1
answer
345
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Compactness of the Fell topology and local compactness
Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I_U = \{A \mid A \cap U \neq \emptyset\}$ ...
- 4,727
1
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1
answer
217
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Connected Hausdorff spaces with different cardinalities of open sets
Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| ...
- 48.9k
1
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1
answer
360
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Holes of a compact set in $\mathbb{R}^n$ that do not contain holes of a larger open set
Let $K$ be a compact subset of an arbitrary open set $\Omega\subset \mathbb{R}^n.$ It is said that a connected component $W$ of $\Omega\setminus K$ is $\Omega$-bounded if $\overline{W}$ is a compact ...
user120667
1
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1
answer
117
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Hausdorff convergence of preimages of discrete-valued functions
Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d_H$ be the Hausdorff ...
- 710
1
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1
answer
454
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Base of topology for metric-like space
Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...
1
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1
answer
166
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Reference for a proof of cancellation property of braid monoids
Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
...
- 6,211
1
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1
answer
245
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Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Let $X$ be a metric space.
In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
- 623
1
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1
answer
97
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Clopen subsets of $P(S)\times {\beta S}$
May be this question turns out trivial, but I can't figure out. I asked on StackExchange, but no one answers.
Let $S$ be a set, or equivalently the topological space with discrete topology and $2$ ...
- 774
1
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1
answer
134
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Chain of interior of closed set
It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such ...
- 11
1
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1
answer
604
views
Partition of Real Number [closed]
Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
- 275
1
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1
answer
83
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Example of a collection of metacompact spaces with non-metacompact box-product
Is there an example of a family $(X_i)_{i\in I}$ of metacompact spaces, such that their box product $\prod_{i\in I}^{\textrm{Box}}X_i$ is not metacompact?
- 48.9k
1
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1
answer
164
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Connectedness of the complements of the connected subsets
EDIT: My original foolish version was instantly destroyed by Dylan Thurston; it consisted of questions 1 & 2 below. Thus now only new question 0 remains to be answered.
Let $\ X:=M^n\ $ be a ...
- 7,500
1
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1
answer
213
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Questions about the dimension-and other properties-of a non-separable topological space
Let the topological space X be the so-called "long line" — which is an uncountable linearly ordered set containing all the countable ordinal numbers as well as a copy of the open unit interval (...
- 6,215
1
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1
answer
686
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dual space of the quotient space of some locally convex topological space
I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
- 1,835
1
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1
answer
371
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countably normed spaces and countably normed spaces [closed]
Why locally convex spaces are not presented as countably normed spaces i.e an infinite sequence of norms (see Generalized functions Tome 2 by Gelfand and Chilov) in the western mathematical ...
- 308
1
vote
1
answer
133
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Special finite subcover of a compact
Let $(a,b)\in \mathbb R^n$. We consider the following open cover of the compact line segment $[a,b]$: $$[a,b]\subset\underset{x\in [a,b]}{\bigcup}B(x,\rho_x),$$
where for $x\in K,B(x,\rho_x)$ is a ...
1
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1
answer
581
views
T2 ⇒ KC ⇒ US ⇒ T1
In a topological KC-space, every compact space is closed.
In a US-space, each convergent sequence has a unique limit.
So, T2 ⇒ KC ⇒ US ⇒ T1, but the converse implications do not hold.
(a): Can ...
- 147
1
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1
answer
104
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Is the Sorgenfrey Line monotonically monolithic?
Just as the title explains, is the Sorgenfrey Line monotonically monolithic (see the definition)?
- 654
1
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2
answers
2k
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General criteria for exhaustion by compact sets
Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
- 113
1
vote
2
answers
660
views
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 ...
- 11
1
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1
answer
334
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topological equivalence of ODEs
Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be Lipschitz.
Define the relation $\stackrel{f}{\sim}$ on $\...
user5810
1
vote
2
answers
341
views
A question about connectedness in Euclidean space [closed]
Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any ...
- 23
1
vote
1
answer
333
views
Do outer regular outer measures always measure open sets?
Let $ \; \langle X,\mathcal{T} \hspace{.06 in} \rangle \; $ be a second-countable Hausdorff space.
Let $ \; \phi : 2^X \to [0,+\infty] \; $ be an outer regular outer measure.
Does it follow ...
user5810
1
vote
1
answer
515
views
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 ...
- 4,829
1
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1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
1
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2
answers
504
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Do all graphs of C1 functions have Hausdorff dimension 1?
Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
1
vote
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,646
1
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2
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193
views
Something like Yoneda's lemma
This is inspired by The Whitehead for maps question.
Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) ...
- 15.1k
1
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4
answers
5k
views
1
vote
2
answers
127
views
Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
- 3,317
1
vote
1
answer
92
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
1
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1
answer
132
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Is the product of torus and sphere a cover of the symmetric square of torus?
Let $T$ denote the $2$-dimensional torus and $T^{(2)}$ denote its symmetric square (which is the orbit space of the canonical $\mathbb{Z}_2$ action on the $4$-torus $T \times T$).
One can see $T^{(2)}$...
- 31
1
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1
answer
104
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Generalizations of Michael theorem
In [1] Michael proved the following:
Theorem. Let $f\colon X \to Y$ be continuous, closed, and onto, where $X$ is $T_1$. If $y \in Y$ is a q-point, then every continuous, real-valued function on $X$ ...
- 115
1
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1
answer
169
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Constructing a continuous function with a prescribed preimage
Given a topological space $X$ and a Banach space $V$, I wonder for which open sets $U$ it is possible to construct a continuous function $f: X \to V$ such that $f^{-1}[B(0, 1)] = U$ - or maybe there ...
1
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1
answer
98
views
Intersection of (relativized/preimage) measure 0 with every hyperarithmetic perfect set
Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual ...
- 3,029
1
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1
answer
177
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Identifying a curve on a closed surface of genus 4
The notation is the one used in the attached picture.
Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
- 66.3k
1
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1
answer
161
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Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
1
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1
answer
105
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abstract description of the topology on a real vector space defined by the algebraically open sets
Let $V$ be a real vector space. Given a subset $A \subseteq V$, say that a point $x \in A$ lies in the algebraic interior of $A$ if every affine line $\ell$ that passes through $x$ has the property ...
- 331
1
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1
answer
111
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Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$,
$$
x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
- 657
1
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1
answer
114
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Continuous surjection between spectra of commutative von Neumann algebras
Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...
- 9,340
1
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1
answer
193
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Is the image of a constructible set between Jacobson spaces constructible if the map takes closed points to closed points?
Let $X$ and $Y$ be two spectral Jacobson spaces and let $f: X \to Y$ be a spectral morphism, i.e. $f$ is continuous and the inverse image of a quasi-compact open is quasi-compact. Suppose further that ...
1
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1
answer
175
views
Does fiber bundles admits good properties of covering spaces?
Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there ...
- 585
1
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3
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345
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Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...
- 6,853
1
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1
answer
195
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Can one explore a surface along ‘piecewise planar’ curves?
Suppose $d\in \{3,4,\dotsc\}$ and $A\subseteq \mathbb{R}^d$ is non-empty, open and connected with its complement $A^c$ connected too and $\text{int}(A^c)\neq \emptyset$. Its boundary $S:=\partial A$ ...
- 1,461
1
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1
answer
190
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Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
user335418
1
vote
1
answer
235
views
Group structure on the strip
Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits_{l\in \{a,b\}}\{d(x,l)\}$.
Let $(X,*)$ be a topological group with the following ...
- 107
1
vote
1
answer
110
views
Existence of a Hölder homeomorphism satisfying prescribed norm constraints
Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...
1
vote
1
answer
188
views
Showing that $C(X,Y)$ is separable when $X$ is compact Hausdorff but $Y$ is just a separable Frechet space
Let $X$ be a compact subset of the Euclidean space. Also let $Y$ be a separable Frechet space with the seminorms $\lVert \cdot \rVert_n$'s.
Then let $C(X,Y)$ be the space of continuous mappings from $...
- 3,477