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Results tagged with metric-spaces
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user 1946
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
12
votes
Accepted
Axiom of Choice and continuous functions
It seems to me that this is provable without using the axiom of
choice.
Suppose that $X$ is a compact metric space and $f:X\to\mathbb{R}$
is continuous. Let's show it is uniformly continuous. Fix any …
- 236k
11
votes
Accepted
Measure of the support of a Borel probability on a metric space
Following Pietro's lead, let me observe that if there is a
measurable cardinal, then there is a counterexample.
Suppose that $\kappa$ is a measurable cardinal. Then there
is a $\kappa$-additive 2-val …
- 236k
7
votes
Which metric spaces have this superposition property?
A very natural concept arises if you should insist that $A$ and
$B$ are small in some way, such as insisting that they are finite.
For example, the countable random graph under the shortest-path metri …
- 236k
5
votes
Are all homogeneous metric spaces bihomogeneous?
Here is a one-dimensional analogue of Richard's triangle
example, obtaining a counterexample in the set of
reals. Namely, replace every integer $n$ with two numbers at
fixed small distance $n\pm\epsil …
- 236k
4
votes
Distance between two points using triangulation
It seems to me that in the general setting of a metric space, what one learns from the sampling data will be precisely the bounds provided by the instances of the triangle inequality that must be obey …
- 236k
3
votes
Accepted
Extending a partially defined metric on a metrizable space
Here is a counterexample to Q2, with your stated extra condition.
Let $X$ consist of the half-open unit interval $(0,1]$ on the $x$-axis in the plane, together with the full unit interval $[0,1]$ at h …
- 236k