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A homeomorphism is a bijective continuous function with a continuous inverse between two topological spaces. It is a 'topological isomorphism' that preserves the topological properties.

The subspace topology on a subset of a topological space is the collection of open sets that are intersections of the subset with open sets of the parent space.

A path-connected space is one in which any two points can be connected by a continuous path within the space.

A local homeomorphism is a function between two topological spaces that is a homeomorphism when restricted to small enough neighborhoods of each point.

A Lindelöf space is a topological space in which every open cover has a countable subcover.

A second countable space is a topological space that has a countable basis for its topology.

An interior point is a point that has a neighborhood completely contained within the set.

A function between two topological spaces is continuous if the preimage of every open set in the codomain is open in the domain.

A complete space is a metric space in which every Cauchy sequence converges to a limit that is within the space.

The product topology on a product of spaces is the coarsest (or weakest) topology that makes all projections continuous.

A limit point of a set is a point that can be approached by points within that set arbitrarily closely, but the point itself does not necessarily have to be in the set.

A neighborhood of a point is a set containing an open set around that point.

The Baire category theorem states that for any complete metric space, the intersection of countably many dense open sets is dense.

An open set is a set that, for every point within it, there exists a neighborhood that is entirely contained within the set.

A closed set is a set that contains all of its limit points; it is also the complement of an open set in some larger space.

A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods. It satisfies the T2 separation axiom.

A boundary point is a point where every neighborhood around it contains at least one point in the set and at least one point not in the set.

A basis for a topology on a set is a collection of sets from which every open set can be formed as a union.

A normal space is a topological space in which any two disjoint closed sets can be separated by disjoint open neighborhoods. It satisfies the T4 separation axiom.

A compact set is a set in which every open cover has a finite subcover. This often implies boundedness and closedness in Euclidean spaces.

A dense set within a topological space is a set whose closure is the entire space.

A metric space is a set equipped with a metric (distance function) that satisfies the conditions of non-negativity, identity of indiscernibles, symmetry, and triangle inequality.

A covering space is a space that 'covers' another space such that each point in the covered space has a neighborhood that is evenly covered by the covering space.

A connected set is a set that cannot be partitioned into two nonempty open sets that do not intersect.