topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
What is known as the Banach-Tarski paradox is the theorem by Banach & Tarski 1924 saying that the axiom of choice implies that any two bounded subsets in Euclidean space of dimension $d \geq 3$ may be partitioned by a finite number of pairwise congruent subsets.
This is perceived as a paradox due to its counter-intuitive interpretation, which becomes particularly vivid if one takes the second bounded subset to be the disjoint union of two copies of the first one: In this case the theorem says, intuitively, that it is possible to break up any shape in 3d Euclidean space into a finite number of pieces, such that re-assembling these pieces suitably yields not just the original shape, but two complete copies of it.
It has been pointed out that it is not just the use of the axiom of choice that is responsible for this perceived paradox, but also the point-based concept of topological spaces as such, see the discussion In point-free topology below.
It is argued by Simpson 2012 that the Banach-Tarski paradox disappears if one works in point-free topology, hence with locales instead of just topological spaces:
We view spaces of interest as locales, and the notion of “part” is given by the standard notion of sublocale, $[\cdots]$. Every topological space determines a locale $[\cdots]$. However, when a space is viewed as a locale, the notion of sublocale gives rise to new “parts” of spaces that are not merely subsets, and need not be determined by their points.
The usual contradictions are avoided $[$this way$]$. The different pieces in the partitions defined by Vitali and by Banach and Tarski are deeply intertangled with each other. According to our notion of “part”, two such intertangled pieces are not disjoint from each other, so additivity does not apply. An intuitive explanation for the failure of disjointness is that, although two such pieces share no point in common, they nevertheless overlap on the topological “glue” that bonds neighbouring points together.
The original article:
Textbook account:
See also:
Discussion in point-free topology:
Last revised on March 12, 2024 at 16:13:28. See the history of this page for a list of all contributions to it.