quaternionic projective line$\,\mathbb{H}P^1$
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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higher geometry / derived geometry
Ingredients
Concepts
geometric little (β,1)-toposes
geometric big (β,1)-toposes
Constructions
fundamental β-groupoid in a locally β-connected (β,1)-topos / of a locally β-connected (β,1)-topos
Examples
derived smooth geometry
Theorems
Given a metric space $(X,d)$ and a point $x \in X$, then the unit sphere $S_x(X) \subset X$ is the subset of those points with unit distance from $x$:
In the Euclidean space $(X,d) = E^n$ of dimension $n$, the unit sphere is the usual (n-1)-sphere $S^{n-1} \simeq S_0(\mathbb{R}^n)$. For $n = 2$ this is the unit circle, for $n = 3$ the unit 2-sphere and so on.
Last revised on December 1, 2019 at 19:09:47. See the history of this page for a list of all contributions to it.