All Questions
12 questions
4
votes
1
answer
279
views
Product topology from two premetric spaces induced by sum of premetrics?
For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.
Do ...
- 463
7
votes
0
answers
493
views
A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
2
votes
0
answers
58
views
A generalization of metrics taking values in partial orders
I'm investigating the origin of the following notion:
Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides).
A $S$...
- 775
4
votes
0
answers
194
views
Are there any major differences in metric topologies and "non-symmetric" metric topologies
Let $X$ be a set and let $d:X\times X\rightarrow [0,\infty)$ satisfy all the axioms of a metric besides symmetry (i.e.: $d$ is a quasi-metric). Define a topology $\tau_{d:+}$ on $X$ induced by $d$ as ...
- 109
8
votes
3
answers
937
views
BCT equivalent to DC
Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
- 143
13
votes
5
answers
1k
views
A generalization of metric spaces
Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....
- 18.6k
5
votes
2
answers
448
views
Space of curves
I am reading Burago, Burago & Ivanov's book where they distinguish the notion of a curve and a path in the following way:
a path in a topological space $X$ is simply a (continuous) map from a ...
- 5,529
2
votes
0
answers
122
views
First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
2
votes
2
answers
241
views
If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 10.5k
4
votes
1
answer
479
views
"monotone" homotopy?
This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and $A\...
- 101
18
votes
1
answer
4k
views
reference for "X compact <=> C_b(X) separable" (X metric space)
I know (and am able to prove via Stone-Čech compactification) that the following is correct:
Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
- 888
8
votes
2
answers
2k
views
End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
- 3,033