A topological space is a set X along with a collection of open subsets T, called a topology on X, such that the empty set and X are in T, the union of any collection of sets in T is in T, and the intersection of a finite collection of sets in T is in T. An example is the set of real numbers R with the standard topology, where open sets are open intervals.

A function f: X → Y between two topological spaces is continuous if the preimage of every open set in Y is an open set in X. This generalizes the notion of continuity in calculus.

A homeomorphism is a continuous function with a continuous inverse between two topological spaces. It is a topological isomorphism, meaning the spaces are "topologically equivalent." An example is the mapping between a circle and an ellipse.

A topological space is connected if it cannot be divided into two disjoint non-empty open sets. If such a division exists, the space is disconnected. An example of a connected space is the real number line R, while two disjoint intervals form a disconnected space.

A topological space is compact if every open cover has a finite subcover. An example of a compact space is the closed interval [0, 1] in the real number line with the standard topology.

A basis for a topology on a set X is a collection of open sets such that every open set in the topology can be written as a union of sets in the basis. For example, the set of all open intervals forms a basis for the standard topology on the real number line.

The product topology on the Cartesian product of two topological spaces X and Y is the topology with basis sets of the form U × V, where U is an open set in X and V is an open set in Y. An example is the product topology on R², where open sets are open rectangles formed from open intervals in R.

A metric space is a set equipped with a distance function (or metric) that satisfies certain properties, such as non-negativity and the triangle inequality. A topological space, on the other hand, is more general and is defined by open sets rather than distances. Every metric space induces a topological space, but not every topological space is a metric space.

Given a topological space X and an equivalence relation ~ on X, the quotient topology on the set of equivalence classes X/~ is defined such that a subset of X/~ is open if and only if its preimage in X is open. An example is the real number line modulo integers, which gives a circle.

A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods. This means that the points can be "separated" by open sets. An example of a Hausdorff space is the real number line with the standard topology.