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A continuous map sends compact subsets of its domain to compact subsets of its codomain.
Let be a continuous map between topological spaces. If is compact, then its image is compact in .
This is one of the most important invariance properties in topology; it underlies results like attainment of maxima and minima on compact domains.
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .
A topological space is a pair , where is a set and is a collection of subsets of , satisfying:
The collection is the topology of the space.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
An image is the set of values a function takes on a subset of its domain: if is a function and , then
The image (often called the range) of is .
The extreme value theorem states: if is a nonempty compact set and is a continuous function, then attains its maximum and minimum values.
That is, there exist such that