nLab
open subspace

Definition

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

For a point-based notion of space such as a topological space, an open subspace is the same thing as an open subset.

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

$$j_U:V\mapsto U\Rightarrow V.$$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 $U$.

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 $\mathrm{Int}(A)$, ${\mathrm{Int}}_X(A)$, $A^{\circ }$, $\stackrel{\circ }{A}$, etc.

(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)

Related concepts

• closed subset, clopen subset

• open cover