All Questions
5,185 questions
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143
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Gluing locally defined continous functions over complex domain
This is a cross-post to the question I asked at MSE over almost a month ago.
Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...
- 195
1
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1
answer
206
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Question on K.Gobel's paper 1969
Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset ...
1
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1
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870
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Borel $\sigma$-algebra on the space of Hölder continuous functions
Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
- 167
1
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1
answer
206
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$S_M$ is not always homeomorphic to the 1-sphere of $F$
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
1
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1
answer
110
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Indecomposable monoids
Let $M$ be a commutative reduced and cancellative monoid and $K(M)$ its group of quotients.
We say that $M$ is indecomposable if for every divisor-closed submonoids $M_1$ and $M_2$, $M=M_1\oplus M_2$...
- 933
1
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1
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158
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Semigroups admitting commutative group actions
Let $(S,*)$ be a semigroup admitting a distinguished element $0$ such that $z*s = s*z = z$, for all $s \in S$. Moreover, let $(\mathbb{G},\cdot)$ be a commutative group. Consider an action
$$
\mathbb{...
- 219
1
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1
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167
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Miscenko example of linearly Lindelof non Lindelof is not normal
In the paper of Norman Howes "A note on transfinite sequences" is mentioned that Miscenko space
$M = \{f \in \prod_{n \in \omega \smallsetminus \{0\}} \aleph_n+1 | \space \exists k \space \space \...
- 11
1
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1
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178
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Existence of a weak Baire space which is not Baire space
A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(...
- 139
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2
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155
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Concerning a strongly nowhere dense subset
A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(...
- 139
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1
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319
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Monoids (or semigroups) with a "finite decomposition" property
In my research I have come across the following condition on a monoid.
Every element $x$ satisfies the following property: there exists a natural number $n$ such that for any $m \geq n$ and any ...
- 408
1
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1
answer
79
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Convergence of a z-filtre to an outer point
Let $X$ be a completely regular topological space and let the set
of all continuous functions from the topological space $X$ into
the topological space $\mathbb{R}$ is denoted by $C(X)$. Let
$Z(X)=\{Z(...
- 11
1
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1
answer
71
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Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
- 1,368
1
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1
answer
175
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Kelley & Namioka's definition of topology of uniform convergence on a subset
In Kelley & Namioka's Linear Topological Spaces, they begin section 8 on Function Spaces with a definition of the topology of uniform convergence. I've reproduced the begining of the first ...
- 323
1
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1
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118
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Almost periodic function and closed spaces
We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$
I think that $X_{2\pi}+...
- 75
1
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1
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134
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Finding necessary and sufficient topological conditions
$\mathcal G: \mathbb R_+ \to \mathbb R_+$ is a set of strictly increasing continuous functions. If for any $\epsilon>0$,$x\in \mathbb R_+$ and $\alpha\in (0,1)$ there exists $z\leq x$ and $g\in \...
- 21
1
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1
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95
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Neighborhoods with proper multiplication
The following question was originally asked here, by C. Dubussy: https://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication
Assume we have two closed subsets $F$ and $...
user75246
1
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1
answer
248
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Tightening a loop
Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
- 109
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1
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160
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Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces
For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...
- 6,215
1
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1
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291
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Topologies for which the ensemble of probability measures is complete
I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess.
...
1
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1
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105
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Open cover not containing a certain subcover
Is there an infinite topological space $(X,\tau)$ with the following property?
There is an open cover ${\cal U}^*$ such that
$X\notin {\cal U}^*$;
every finite subset $F\subseteq X$ is contained in ...
- 48.9k
1
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1
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106
views
Similarity graph for continuous maps between Hausdorff spaces
Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are similar if for all $V\subseteq Y$ open we have either
$f^{-1}(V) = g^{-1}(V) = \emptyset$, or
$f^{-1}(V) \...
- 48.9k
1
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1
answer
254
views
Interval topology and order convergence topology
Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\...
user62017
1
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1
answer
370
views
Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space
This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...
- 191
1
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1
answer
84
views
Number of continuous characters on an infinite Hausdorff precompact abelian group with exponent $p$
Let $(G,\mathcal T)$ be an infinite Hausdorff precompact abelian topological group and let $G$ have exponent $p$ where $p$ is a prime number.
Can it be proved that there are at least $p+1$ continuous ...
- 1,145
1
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1
answer
196
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Lax monoids where only the unit triangle is lax
I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...
- 327
1
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1
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194
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Modern reference request concerning Efimov's "On dyadic spaces"
Is there any modern reference (book, textbook, monograph, etc.) that contains the following result of B. Efimov (On dyadic spaces // Dokl. Akad. Nauk SSSR 151 (1963) (Russian). English translation: ...
- 895
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1
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189
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subspace in pseudotopological space
Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...
- 41
1
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1
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655
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Topological razors (ball-like spaces)
Introduction
Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...
- 7,500
1
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1
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132
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Generalized connected components decomposition for Priestley spaces
Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...
- 2,464
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1
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172
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Normal Uniform Spaces and their function uniform spaces
Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase
$$\Lambda =\{ \{(f,...
user31967
1
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1
answer
171
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Non-compact structure group and compactly supported gauge transformations
Let $\pi\colon P\to X$ be a locally trivial principal $G$-bundle over a Hausdorff paracompact space $X$, where $G$ is a topological group (we work in the category of topological spaces, as I do not ...
- 523
1
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1
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101
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ball in universal cover belongs to the union of actions on a section?
M is an n-dim manifold. $\pi :\tilde M \to M$ the universal cover of M. $\tilde p \in \tilde M$ a lift of p. We choose a measurable section $j:{B_1}\left( p \right) \to {B_1}\left( {\tilde p} \right)$,...
- 689
1
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1
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72
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Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
1
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1
answer
179
views
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 ...
- 723
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1
answer
216
views
Counting modular squares in an interval
For an integer $m$, let $S^m_{x_0,x_1} = \{ t | x_0 ≤ t ≤ x_1 $ and $t$ is a square modulo $m \}$. Let $S^m_x$ = $S^m_{0,x}$.
Determining whether the sets $S^m_x$ are empty is easy (1 is always a ...
- 123
1
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1
answer
148
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Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
- 507
1
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1
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1k
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Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
1
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1
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317
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Mapping class group and cylindrical structure
Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
- 93
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1
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2k
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Covering the Rationals -- A Paradox? [closed]
Covering the Rationals -- A Paradox?
The following seems to yield a paradox. Can anyone provide the proper resolution?
Preamble
It is easy to show that between any two reals there is a rational. If ...
1
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1
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163
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Term for number of crossings of smooth curves
Two smooth oriented finite curves $g_1, g_2$ on e.g. the 2-dimensional torus can intersect each other transversally in two ways: either the pair $(Tg_1(x),Tg_2(x))$ of tangent vectors in the ...
1
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1
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131
views
Conditions under which a given scheme converges
I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set
$\Delta_{n-...
- 645
1
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1
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595
views
When is a bijective map between bundles a homeomorphism?
Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...
- 165
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1
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374
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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 ...
user5810
1
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1
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154
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undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$
Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm ...
- 549
1
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1
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362
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Winding number bijection on graphs
Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...
- 4,952
1
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1
answer
390
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Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 13
1
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1
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909
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
1
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0
answers
133
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
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0
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42
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Why does the Kieboom characterization of shape is restricted only to paracompact spaces?
Borsuk founded shape theory as an extension of homotopy theory, appropriate for spaces with bad local properties. Borsuks definition was applied only to compact metric spaces. Later, this was ...
- 111
1
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0
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104
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Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127