Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
A function from one space to another is closed if the image of every closed set is closed.
A space is completely normal if any two separated sets have disjoint neighbourhoods.
A space is called a continuum if it a compact, connected Hausdorff space.
A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is the whole space.
A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space.
A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure.
A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.
A space is metacompact if every open cover has a point finite open refinement.
A metric invariant is a property which is preserved under isometric isomorphism.
A space is metrizable if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.
A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point".
A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space.
A space is polyadic if it is the continuous image of the power of a one-point compactification of a locally compact, non-compact Hausdorff space.
A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections.
A space is pseudocompact if every real-valued continuous function on the space is bounded.
A space is rim-compact if it has a base of open sets whose boundaries are compact.
A space is sequentially compact if every sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.
A space is simply connected if it is path-connected and every loop is homotopic to a constant map.

Here are some facts about submaximality as a property of topological spaces:

A space is totally disconnected if it has no connected subset with more than one point.
The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.
See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)