nLab open subspace

Idea

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

When ordered by inclusion the open subspaces form a poset, the frame of opens.

Definition

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

For topological spaces

If $X$ 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 $X$).

For convergence spaces

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

For locales

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

$$j_{U}\colon V \mapsto U \Rightarrow V .$$

The idea is that this subspace is the part of $X$ which involves only $U$, and we may identify $V$ with $U \Rightarrow V$ when we are looking only at $U$. If $X$ 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 $U$ (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: $U\subseteq X$ is open if for any $x\in U$ and $y\in X$, either $x\neq y$ or $y\in U$. 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 $A$ of a space $X$ is open if the inclusion map $A \hookrightarrow X$ is an open map.

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

Related concepts

• closed subset, clopen subset

• open point

• open cover

• open map

• overt space

• saturated subset

References

• Jacques Penon, Topologie et intuitionnisme

• Eduardo Dubuc and Jacques Penon, Objets compactes dans les topos