Space: A space is a set of points, along with a set of rules for determining when two points are "close" to each other.

Topological space: A topological space is a set of points equipped with a topology, which is a collection of open sets that define a notion of closeness.

Continuity: A function is continuous if it preserves the topological structure of the space, meaning that the function maps open sets to open sets.

Compactness: A topological space is compact if every open cover has a finite subcover.

Connectedness: A topological space is connected if it cannot be written as the union of two disjoint nonempty open sets.

Separation axioms: A topological space is said to satisfy the separation axioms if it satisfies certain conditions related to the separation of points and sets. For example, a space is said to be Hausdorff if for every pair of distinct points, there are disjoint neighborhoods around each point.

Homotopy: Two continuous functions are homotopic if one can be continuously deformed into the other.