Math and science::Topology

# 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...].

### 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:

1. $$f$$ is continuous.
2. For every subset $$A$$ of $$X$$, one has [$$f(\bar{A}) \subseteq \text{what set?}$$].
3. For every closed set $$B$$ of $$Y$$, the set [...] is [...].
4. For each $$x \in X$$ and each neighbourhood $$V$$ of $$f(x)$$, there is a [...] such that [...].