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Results tagged with metric-spaces
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user 3969
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) $d(x,y)+d(y,z) \geq d(x,z)$.
6
votes
For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
Note that $\mathbb{R}^n\times Q$ is proper homotopy equivalent to $\mathbb{R}^n$.
If they were homeomorphic, we would obtain a proper homotopy equivalence between $\mathbb{R}^m$ and $\mathbb{R}^n$.
Us …
- 11.1k
4
votes
1
answer
241
views
cardinality of local bases in the non-standard reals
Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{R} …
- 11.1k