Continuous maps
Continuous maps
Let \( X \) and \( Y \) be topological spaces. A function \( f : X \to Y \) 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 \( f : X \to Y \) be a continuous map, and let \( (x_n) \) be a sequence in \( X \) converging to \( x \in X \); 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 \( f : X \to Y \) be a function. The the following are equivalent:
- \( f \) is continuous.
- For every subset \( A \) of \( X \), one has [\( f(\bar{A}) \subseteq \text{what set?} \)].
- For every closed set \( B \) of \( Y \), the set [...] is [...].
- For each \( x \in X \) and each neighbourhood \( V \) of \( f(x) \), there is a [...] such that [...].