All Questions
10
questions
16
votes
1
answer
502
views
Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
(cross-posted from this math.SE question)
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
- 665
3
votes
0
answers
77
views
Every Borel linearly independent set has Borel linear hull (reference?)
I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone.
Theorem. The linear hull of any linearly independent Borel set in a Polish ...
- 40.2k
5
votes
1
answer
240
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 2,672
4
votes
0
answers
103
views
Borel selections of usco maps on metrizable compacta
The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
- 40.2k
11
votes
2
answers
973
views
How to show that something is not completely metrizable
I have a Polish space $X$ and a subset $A \subset X$.
I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$.
This means: If I want to show ...
- 967
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
2
votes
0
answers
168
views
Classify spaces that make extension theorems hold
Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...
- 6,167
9
votes
1
answer
557
views
Is $\ell^\infty$ Polishable?
Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
- 2,971
3
votes
1
answer
1k
views
Different Metrics for Baire Space and their induced Topologies
The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ \begin{...
- 798
15
votes
2
answers
3k
views
Generalizations of the Tietze extension theorem (and Lusin's theorem)
I am reasking a year-old math.stackexchange.com question asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
The Tietze ...
- 6,167