New topological spaces from old

The main examples are the following:

Subspaces:

Given a topological space $X$ and a subset $A$ of $X$. Then the subset $A$ becomes a topological space with the relative topology.

Products:

Given two topological spaces $X$ and $Y$. Then the product set $X \times Y$ becomes a topological space with the product topology.

Infinite products are also possible.

Quotients:

Given a topological space $X$ and an equivalence relation $\sim$ on $X$. Then the quotient set $X/ \sim$ becomes a topological space with the quotient topology.

Each of the three produces a continuous function:

• the inclusion $i_A \colon A \rightarrow X$ given by $i_A(x)=x$,
• the projections $p_X \colon X \times Y \rightarrow X$ and $p_Y \colon X \times Y \rightarrow Y$ given by $p_X(x,y) = x, p_Y(x,y) = y$,
• the quotient map $q \colon X \rightarrow X/ \sim$ given by $q(x) = [x]$.

See Examples of maps.

For algebraic topology purposes we create new complexes from old instead.

In the group theory, the analogs are respectively:

• subgroups,
• product groups,
• quotient groups.

See New groups from old.