All Questions
5,184 questions
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Questions on the proof Lemma 4.5 GTM 175, Lickorish
I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.
For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\...
- 15
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1
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129
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Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
...
0
votes
1
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267
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Is this subset of $[0,1]$ of second category?
Let $S$ be an uncountable subset of $[0,1]$ such that:
$S$ is dense in $[0,1]$;
as a topological space, $S$ is Baire.
Is it true that $S$ is of second category as a subset of $[0,1]$?
- 4,459
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votes
2
answers
323
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Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$.
I would like to show that, ...
- 167
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81
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If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?
Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...
- 199
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1
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114
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$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?
Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that
$$
A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0.
$$
Can we say that :
$$
\overline{(A-A)}\cap\overline{...
- 117
0
votes
1
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297
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Mysior's example of not completely Hausdorff space
https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601748-4/S0002-9939-1981-0601748-4.pdf
In this link, there is the example of regular space, that is not completely regular. This space ...
- 73
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164
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Restriction of non-metrizable topology to dense subset is non-metrizable
Let $(X,\tau)$ be a non-metrizable topological space which is not first-countable and let $\emptyset \neq Y\subset X$ be a proper dense subset. Is it possible for $(Y,\tau_Y)$ (where $\tau_Y$ is the ...
- 5,405
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268
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Topologies and Borel $\sigma$-fields on disjoint unions
Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
Consider ...
- 195
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554
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Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
- 4,195
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votes
1
answer
102
views
On a pair of continuous functions "connected" by continuous functions
Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$
...
- 48.9k
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1
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66
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Cardinality of the topology in countable connected $T_2$-spaces
If $(X,\tau)$ is a connected $T_2$-space with $|X|=\aleph_0$, what values can $|\tau|$ take?
- 48.9k
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655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
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142
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A property of compact topological space via certain $C^*$ embedding in operator algebras
Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$?
If not, is the answer affirmative when $A$ is ...
- 356
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1
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153
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C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
0
votes
1
answer
68
views
Connected $T_2$-spaces with nowhere dense covering number $3$
This is a special case of a question that has not been answered so far.
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the ...
- 48.9k
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1
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127
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Product topology and order convergence topology
Let $(P,\leq)$ be a poset. We define the order convergence topology, denoted by $\tau_o(P)$. By a set filter $\mathcal{F}$ on $P$ we mean a collection of subsets of $P$ such that:
$\emptyset \notin \...
- 48.9k
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1
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82
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Maximal discrete subsets of connected $T_2$-spaces
If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.
Is there an infinite connected $T_2$-space $(X,\tau)$ ...
- 48.9k
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55
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On 1-iso maps and subsets of the unit circle
Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
- 97
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843
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$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
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482
views
Complement of a finite union of convex sets
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...
- 2,933
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1
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153
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Path-connected Hausdorff interval topologies
Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$?
(This is a follow-up ...
- 48.9k
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1
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381
views
Maximal group image
How does one prove: if $S$ is a finitely generated Clifford semigroup its maximal group image is actually $S_{e_{n}}$?
- 1
0
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1
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149
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Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?
Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...
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1
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91
views
Intersection of complements of connected components
Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^*$. Let ${\cal C}$ be the ...
- 48.9k
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1
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107
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Topology : Study on Separation Properties [closed]
I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
- 115
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1
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182
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Rational points in the Alexandroff line
Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line.
Is $X$ a metrizable space?
- 356
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1
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111
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Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
- 135
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1
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425
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Property of Mrowka
A topological space $X$ satisfies Property $K_1$ (Property of Mrowka) if the closure of the union of arbitrarily many $G_\delta$ sets of $X$ coincides with its sequential closure (the sequential ...
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1
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208
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The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
- 9
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1
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388
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Does there exist a topology for a set X which is compact and Hausdorff? [closed]
For every set $X$ and every topology $\tau$ over $X$ we have that $\tau$ contains the trivial topology $\{ X, \emptyset\}$, which is compact, and is contained in the discrete topology $\{ S: S \...
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2
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259
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Can we generalize the result of Urysohn's lemma to countable collection of pairwise disjoint closed subsets of a normal space..?
Suppose X is a normal topological space. Suppose some metric space for example.
If {$A_n$}$_{n=1}^{\infty}$ is a collection of pairwise disjoint closed subsets of X, can we find a continuous function ...
- 111
0
votes
2
answers
210
views
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 ...
- 1,704
0
votes
2
answers
359
views
some questions on Lindelöf property
I have several questions on Lindelöf property.
If every point countable open cover of $X$ has a countable subcover (Condition A), does $X$ have Lindelöf property? How far is having Condition A from ...
- 654
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1
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209
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On generalized ordered spaces
Let X be a Go space. If G is open in X, why is every convex component of G open?
( It is well known that any non-void subset G of X can be uniquely represented as a union
of its maximal convex ...
- 654
0
votes
1
answer
296
views
homeomorphism of topological group
Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ ...
- 1
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1
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285
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A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
0
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1
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319
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Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
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1
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322
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Topological Properties of Non-Smooth Functions [closed]
I am interested to know what topological properties are possessed by non-smooth functions. I suspect this is well known, but I can't find what I am looking for from google.
Is there a topological "...
- 137
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1
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145
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Can we describe open cover compactness of a space in how the space relates to other spaces?
I've seen two definitions of connectedness of categorical flavour which I present below:
(Maps into two point set): A topological space $X$ is connected iff the only continous functions $f:X \to \{ 0,...
- 1,535
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1
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142
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"Locally compact"-ly generated topological spaces
Let $P$ be a property of topological spaces - here I am interested in "compact" and "locally compact".
A topological space $X$ is $P$-ly-generated if, for any topological space $Y$,...
user531303
0
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1
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78
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Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]
Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components.
Let $U$ be a connected component of $S \setminus K$ and ...
0
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1
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96
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A question about filterbasis
K. Hardy and R. G. Wood assert that the family in line 4 is a filterbase. I couldn't show it.
- 1,367
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1
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188
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A question about uniformities generated by pseudometrics
Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X
$. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left(
a_{n}\right) $ is a sequence of positive ...
- 1,367
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1
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92
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Property stronger than $T_1$ and weaker than regularity
Recently I got interested in the following property of topological spaces:
$(X,\mathcal{T})$ satisfies (P) if the following holds:
For any nonempty closed subsets $F$ and $G$ with $F\ne G$, there are ...
0
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1
answer
144
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Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 73
0
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1
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291
views
Tensor product is complete?
Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
- 177
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1
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243
views
Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]
Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
- 839
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1
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159
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Partial orders on downward closed sets [closed]
Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; ...
- 639
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1
answer
43
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Exhaustions of product subsets by smaller product subsets
Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences ...
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