All Questions
Tagged with gn.general-topology dimension-theory
64 questions
1
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2
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202
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Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 1,201
2
votes
1
answer
103
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LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
I am looking for locally compact Hausdorff spaces $X$ with the following property:
If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional.
One can see ...
- 1,201
11
votes
2
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314
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Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,201
0
votes
1
answer
79
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Dimension of a manifold derived from a dense $G_{\delta}$ subspace
Let $X,Y$ be (compact connected) topological manifolds of dimensions $n,m$, respectively. Assume that a dense $G_{\delta}$ subspace $A$ of $X$ is homeomorphic to a dense $G_{\delta}$ subspace $B$ of $...
1
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1
answer
130
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A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are dense and contractible
Inspired by this question we ask the following question.
Note that the comment conversation and answers to the above question imply that
There are two complementary subsets of the unit ...
- 356
3
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1
answer
621
views
What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...
- 612
3
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1
answer
226
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$\sigma$-product of the Hilbert cube
Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$
("eventually&...
- 3,317
3
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0
answers
69
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Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
3
votes
0
answers
107
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Does the pseudo-arc contain Erdős space?
The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense ...
- 3,317
7
votes
1
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262
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Can you remove a zero dimensional subspace from a cube and obtain a planar space?
The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title:
Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be ...
- 3,011
5
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1
answer
198
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Iterating the dimensional kernel of a metric space
Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let
\begin{align}
\Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\
\Lambda^2(X)&=\Lambda(\...
- 3,317
6
votes
0
answers
111
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A generalized Hausdorff dimension in form of a Lower semi continuous function
Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
- 356
4
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0
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115
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Dimension properties of some concrete hereditarily disconnected subspaces of the Hilbert cube
This question was motivated by this MO-question asking about the example of a hereditarily disconnected metrizable separable space, which is not the union of countably many totally disconnected ...
- 41.8k
6
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0
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144
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Is there a hereditarily disconnected space which is not the union of countably many totally disconnected subspaces?
A topological space $X$ is called
$\bullet$ totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$;
$\bullet$ ...
- 41.8k
6
votes
1
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184
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Classification of Polish spaces up to a $\sigma$-homeomorphism
A function $f:X\to Y$ between topological spaces is called
$\bullet$ $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
- 41.8k
10
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0
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173
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Is there a universal totally disconnected Polish space?
A topological space $X$ is called totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$.
In 1973 Roman Pol proved that ...
- 41.8k
4
votes
1
answer
133
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Sufficient conditions for the covering dimension and large inductive dimension of compact Hausdorff spaces to coincide
I have been looking through Alan Pears' "Dimension theory of general spaces" recently. In this book Pears references a 1960 paper by Aleksandrov and Ponomarev called "Some classes of $n$...
- 213
38
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1
answer
1k
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Sequences with 0's in $\mathbb R ^\omega$
Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of ...
- 3,317
2
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0
answers
165
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Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this ...
- 1,648
3
votes
1
answer
266
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Embedding CW-complexes into infinite-dimensional topological vector spaces
Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
- 519
2
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0
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83
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Increasing a nowhere dense set in $\mathfrak E_{\mathrm{c}}$
Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$
Can you always find a closed nowhere ...
- 3,317
3
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0
answers
81
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Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$
On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
- 439
6
votes
0
answers
107
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Existence of stable spaces
An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...
- 3,317
5
votes
0
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182
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Which metrizable spaces can be embedded into the countable power of $\omega$ with the cofinite topology?
Let $\omega_{cf}$ be the countable space $\omega=\{0,1,2,3,\dots\}$ endowed with the cofinite topology $$\tau_{cf}=\{\emptyset\}\cup\{U\subseteq\omega:\omega\setminus U\mbox{ is finite}\}.$$
It is ...
- 41.8k
5
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1
answer
291
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Is the Hilbert cube the countable union of punctiform spaces?
Recall that a (separable) metric space is called punctiform, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves ...
- 4,717
9
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2
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505
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A natural $\mathbb Q\times \mathbb P$ subset of $\mathbb R$?
I would like a simple description of a dense subset of $\mathbb R$ which is homeomorphic to $\mathbb Q\times \mathbb P$. Preferably the description will be of an algebraic nature, and perhaps the set ...
- 3,317
1
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1
answer
132
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Perfect images of complete Erdős space
Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact ...
- 3,317
1
vote
1
answer
234
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Quotients of the irrationals
Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
- 3,317
4
votes
0
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441
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The "core" of complete Erdős space
This question is about the Erdős spaces:
$\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and
$\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\}...
- 3,317
10
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2
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871
views
Is the complement of a zero-dimensional subset of the plane path-connected?
Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $...
- 3,317
6
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1
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231
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A classification of $G_{\delta\sigma}$ zero-dimensional spaces?
Among separable metrizable spaces:
Cantor set is the unique compact zero-dimensional space without isolated points.
$\mathbb Q$ is the unique countable space without isolated points
$\mathbb R \...
- 3,317
4
votes
2
answers
1k
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A set whose Hausdorff dimension gradually changes?
Can there be a set whose Hausdorff dimension gradually changes?
For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...
- 10.1k
11
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1
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992
views
Why are homeomorphism groups important?
For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
- 3,317
3
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0
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146
views
Separating a countable closed set from a point
Let 𝑋 be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set.
Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\...
- 3,317
6
votes
1
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489
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Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional
A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$.
A ...
- 4,535
5
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1
answer
213
views
One-dimensional compacta as projective limits
Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1.
Is it possible to express $X$ as a filtering projective limit of finite graphs?
Here finite graphs ...
- 53
3
votes
1
answer
2k
views
Dimension of a topological space equals the supremum of the dimension of open subsets in an open cover
For a topological space $X$ which is covered by a family of open subsets $\{U_i\}$, show that $\dim(X)=\sup (\dim(U_i))$.
I understand that $\dim(X)\geq \sup(\dim(U_i))$, so it only suffices to show ...
- 41
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3
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1k
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Topological dimension of the image of continuous surjective functions
Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$.
Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...
- 71
32
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1
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1k
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If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?
I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
- 439
1
vote
1
answer
260
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Understanding equivalent condition for covering dimension
Let dim $X$ denote the Lebesgue covering dimension for a topological space $X$. Now a result in common books concerning dimension theory states the following:
If $X$ is a normal topological space, ...
- 149
2
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1
answer
133
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Topologically Ordered Families of Disjoint Cantor Sets in $I$?
Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
- 439
3
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2
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152
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What is the dimension of a subspace of the product of $n$ linearly ordered compacta
This question is motivated by this problem of Dominic van der Zypen.
Problem. Let $X=\prod_{i=1}^nX_i$ be the Tychonoff product of linearly ordered compact Hausdorff spaces $X_1,\dots,X_n$. Is it ...
- 41.8k
17
votes
1
answer
525
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Lowest Dimension for Counterexample in Topological Manifold Factorization
Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
- 439
6
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1
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422
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Transitive homeomorphisms of Erdős spaces
A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...
- 3,317
5
votes
1
answer
209
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Direct limit of Cantor sets
Let $C$ be the Cantor set, and $\omega$ the discrete space of integers $\{0,1,2,...\}$.
My conjecture:
(1) For each $n<\omega$ let $f_n:C\to C$ be a continuous function (possibly not onto). Let ...
- 955
7
votes
2
answers
546
views
Doubling dimension vs other metric dimensions
For separable metric spaces, three fundamental notions of dimension
are equivalent:
$$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$
Where does the doubling dimension
fit into the picture?
- 6,541
3
votes
1
answer
148
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Fiber dimension formula for compact Hausdorff spaces?
In Algebraic Geometry one has a very useful formula for the dimension of fibers. Specifically I am thinking about a statement of the following form:
Let $C$ be a curve over $\mathbb{C}$, and let $S$...
- 335
2
votes
1
answer
406
views
Extremally disconnectedness and 0-dimensional space
Let $X$ be a non-empty topological space. Then we have the following concepts for the topological space $X $:
1) We say $X $ has property $*$, if for every closed subset $A$ of $X$ and every open ...
- 21
2
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1
answer
370
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One-dimensional topological spaces
We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the ...
- 95
3
votes
1
answer
160
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Topological spaces with Lebesgue covering dimension 1
We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general
connected topological space with the ...
- 95