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
A topological (or spatial) locale is a locale that comes from a topological space. This is an extra property of locales, a property of having enough points.
Let $X$ be a topological space. Then we may define a locale, denoted $\Omega(X)$, whose frame of opens is precisely the frame of open subspaces of $X$.
A locale is topological, or spatial, if it is isomorphic to $\Omega(X)$ for some topological space $X$.
A locale $L$ has enough points if, given any two opens $U$ and $V$ in $L$, $U = V$ if (hence iff) precisely the same points of $L$ belong to $U$ as belong to $V$.
The following conditions are all logically equivalent on a locale $L$:
(It would be nice to state this as a theorem and put in a proof.)
Basically, what is going on here is that we have an idempotent adjunction from topological spaces to locales, and the topological locales comprise the image of this adjunction. The corresponding condition on topological spaces is being sober.
Therefore, the full subcategory of $Loc$ on the topological locales is equivalent to the full subcategory of $Top$ on sober spaces.
The terms ‘topological locale’ and ‘spatial locale’ can be confusing; they suggest a locale in Top or in some category Sp of spaces, which is not correct. Instead, the adjective ‘topological’ and ‘spatial’ should be taken in the same vein as ‘localic’ in ‘localic topos’ or ‘topological’ in ‘topological convergence’. These two terms also suggest that the study of other locales is not part of topology or that these other locales are not spaces, which is also incorrect.
The really clear term for a topological locale is ‘locale with enough points to separate the opens’, but ‘locale with enough points’ should be unambiguous. However, it is still a bit long. The shortest term, ‘spatial locale’, is probably also the most common. Occasionally one sees ‘spacial’ instead of ‘spatial’, but this might just be a misspelling.
Assuming the axiom of choice, locally compact locales are spatial. In particular, compact regular locales are locally compact, hence automatically spatial. Coherent locales? are also spatial.
More generally, the meet of a countable family of open sublocales (i.e., a $G_\delta$-sublocale) of a compact regular locale is spatial.
The completion of a uniform locale with a countable basis of uniformity is spatial.
Last revised on July 25, 2019 at 11:19:56. See the history of this page for a list of all contributions to it.