0
votes
0answers
12 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
2
votes
1answer
53 views

What makes metric spaces special?

This is not a question about what is special about a metric space in itself; instead, I'm wondering what sets metric spaces apart from uniform spaces? An explanation is in order. As a parallel, when ...
2
votes
1answer
32 views

The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
12
votes
1answer
155 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
4
votes
1answer
164 views

Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
1
vote
2answers
60 views

Open sets in a topology produced by metrics.

For each $i\in I$, $$d_i:X^2\to \Bbb R$$ is a metric and $\mathcal T$ is the coarsest topology on $X$ containing all topologies produced by the $d_i$ . For $U\subseteq X$ we have: $$\forall x \in ...
1
vote
1answer
79 views

a counterexample for Uniform Spaces

Uniform Space is a generalization of metric spaces . In a uniform space the closure of a singleton $\{x\}$ is the intersection of all neighborhoods of $x$. Find an infinite topological space such ...
10
votes
1answer
248 views

Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...