nLab
open subspace

Contents

Idea

An open subspace UU of a space XX is a subspace of XX whose elements are “stable to small perturbations”, or can be “observed to belong to UU by measurements with finite precision”.

Definition

Depending on what sort of “space” XX is, this may be defined in different ways.

For topological spaces

If XX 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 XX).

For convergence spaces

If XX is a convergence space (or some variation such as a subsequential space), we define a subset UXU\subseteq X to be open if any filter/net/sequence converging to a point of UU must be eventually in UU. This defines an “underlying topological space” of XX.

For locales

In locale theory, every open UU in the locale defines an “open sublocale” which is given by the open nucleus

j U:VUV. j_{U}\colon V \mapsto U \Rightarrow V .

The idea is that this subspace is the part of XX which involves only UU, and we may identify VV with UVU \Rightarrow V when we are looking only at UU. If XX 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 UU (and this is true even in constructive mathematics).

For toposes

See open subtopos.

In synthetic topology

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: UXU\subseteq X is open if for any xUx\in U and yXy\in X, either xyx\neq y or yUy\in U. 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.

Properties

A subspace AA of a space XX is open if the inclusion map AXA \hookrightarrow X is an open map.

The interior of any subspace AA is the largest open subspace contained in AA, that is the union of all open subspaces of AA. The interior of AA is variously denoted Int(A)Int(A), Int X(A)Int_X(A), A A^\circ, A\overset{\circ}A, etc.

References

Revised on November 30, 2016 16:14:47 by Mike Shulman (76.167.222.204)