Continuous maps
Continuous maps
Let
In short, continuity means that [phrased in a few simple words...].
Some results
Continuous maps preserve convergence of sequences.
Let
In metric spaces this lemma is an if and only if statement, whereas for topological spaces we are restricted to only the forward implication above; it is possible to construct discontinuous maps of topological spaces that, nevertheless, preserve convergence of sequences.
The composite of continuous maps [is always continuous/need not be continuous?].
The inverse of a continuous bijection [is always continuous/need not be continuous?].
Munkres presents three statements that are equivalent to stating that a function is continuous:
Continuity equivalences
Let
is continuous.- For every subset
of , one has [ ]. - For every closed set
of , the set [...] is [...]. - For each
and each neighbourhood of , there is a [...] such that [...].