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
An open subspace of a space is a subspace of whose elements are “stable to small perturbations”, or can be “observed to belong to by measurements with finite precision”.
When ordered by inclusion the open subspaces form a poset, the frame of opens.
Depending on what sort of “space” is, this may be defined in different ways.
If is a topological space in the classical sense, then it is defined by specifying a collection of subsets to be called “open” (that are closed under finite intersections and arbitrary joins, i.e. are a sub-frame of the power set of ).
If is a convergence space (or some variation such as a subsequential space), we define a subset to be open if any filter/net/sequence converging to a point of must be eventually in . This defines an “underlying topological space” of .
In locale theory, every open in the locale defines an “open sublocale” which is given by the open nucleus
The idea is that this subspace is the part of which involves only , and we may identify with when we are looking only at . If is a (sober) topological space regarded as a locale, then any such open sublocale is spatial and coincides with the subspace determined by the subset (and this is true even in constructive mathematics).
See open subtopos.
In synthetic topology, we interpret ‘space’ as meaning simply ‘set’ (or type, i.e. the basic objects of our foundational system). There are then multiple ways to define “open subset”.
One is to use a dominance, which specifies the open subsets representably by a set of “open truth values”.
Another possibility is the following definition due to Penon: is open if for any and , either or . This definition does not require a choice of dominance, but it is generally only correct for Hausdorff spaces; for instance, the open point in the Sierpinski space is not open in this sense. However, in various gros toposes of topological, smooth, or algebraic spaces it does induce the correct notion of open subsets for Hausdorff spaces; see Dubuc-Penon.
A subspace of a space is open if the inclusion map is an open map.
The interior of any subspace is the largest open subspace contained in , that is the union of all open subspaces of . The interior of is variously denoted , , , , etc.
Jacques Penon, Topologie et intuitionnisme
Eduardo Dubuc and Jacques Penon, Objets compactes dans les topos
Last revised on August 28, 2024 at 11:18:28. See the history of this page for a list of all contributions to it.