CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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
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 $Int(A)$, $Int_X(A)$, $A^\circ$, $\overset{\circ}A$, etc.
(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)