2
votes
0answers
143 views

Is there a better function (linear or even a projection)?

Let $A$ be a finite $n$-element set. Let $\mathbb R^A$ be an $n$-dimensional Euclidean space (with the ordinary Euclidean distance). Let $X$ be an arbitrary topological space. Consider a continuous ...
1
vote
3answers
761 views

Zero-dimensional space

Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that ...
15
votes
2answers
486 views

A “dimension” for Tychonoff spaces

It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...
0
votes
0answers
176 views

Decompose a set into sets of Hausdorff-dimension n-1

Assume we can decompose a set $A$ in $\mathbb{R^n}$ of Hausdorff-dimension n into sets $(A_t)$ $t\in [0,1]$ of Hausdorff-dimension n-1 whose n-1-dimensional volume is known (for example is zero). ...
1
vote
0answers
168 views

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 ...
4
votes
1answer
676 views

Topological dimension, is it local?

Let $n\in\mathbb N$ and $X$ be a complete metric space. Assume that there is $\epsilon>0$ such that $$\dim B_\epsilon(x)\le n$$ for any $x\in X$. Is it true that $\dim X\le n$? Here ...
16
votes
3answers
3k views

Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$? The question closely related to ...
10
votes
1answer
389 views

Can dividing out a group action can increase the Lebesgue dimension ?

Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$? I know examples, ...
4
votes
1answer
252 views

Lebesgue dimension of closures satisfying the Z-set condition

Given any subspace $A\subset X$ of a topological space with Lebesgue dimension $\le N$. Let $\bar{A}$ denote the closure of $A$. Assume, that the pair $(\bar{A},A)$ satisfies the Z-set condition, ...
0
votes
3answers
206 views

how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$: $\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$, where $\sim$ denotes ...