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New topological spaces from old
From Mathematics Is A Science
Revision as of 20:20, 21 July 2011 by imported>Tom
The main examples are the following:
Subspaces:
Products:
Infinite products are also possible.
Quotients:
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:
See New groups from old.