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Math and science::Topology

Continuous maps

Continuous maps

Let X and Y be topological spaces. A function f:XY is continuous iff [for every something, that something has some property].

In short, continuity means that [phrased in a few simple words...].

Some results

Continuous maps preserve convergence of sequences.

Let f:XY be a continuous map, and let (xn) be a sequence in X converging to xX; then the sequence [...] converges to [...].

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 X and Y be topological spaces and let f:XY be a function. The the following are equivalent:

  1. f is continuous.
  2. For every subset A of X, one has [f(A¯)what set?].
  3. For every closed set B of Y, the set [...] is [...].
  4. For each xX and each neighbourhood V of f(x), there is a [...] such that [...].