What is constructions?

Three ways to make new spaces from old:

1. Subspace: on $A \subseteq X$, the open sets are the $U \cap A$, $U$ open in $X$ — the coarsest topology making the inclusion continuous.
2. Product: on $X \times Y$ (and finite products), the topology with basis the open boxes $U \times V$; on an infinite product $\prod_i X_i$, the basis consists of boxes $\prod U_i$ with $U_i = X_i$ for all but finitely many $i$ — the coarsest topology making every projection continuous.
3. Quotient: if $\sim$ is an equivalence on $X$ and $\pi\colon X \to X/{\sim}$ the projection, declare $V \subseteq X/{\sim}$ open iff $\pi^{-1}(V)$ is open — the finest topology making $\pi$ continuous.