the trivial topology. Compactness Compact. A space is compact if every open cover has a finite subcover. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal. Sequentially compact. A space is sequentially compact if every sequence has a convergent subsequence. Countably compact. A space is countably compact if every countable open cover has a finite subcover. Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded. σ-compact. A space is σ-compact if it is the union of countably many compact subsets. Lindelöf. A space is Lindelöf if every open cover has a countable subcover. Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal. Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff. Ultraconnected compact. In an ultra-connected compact space X every open cover must contain X itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith. Metrizability Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T. Polish. A space is called Polish if it is metrizable with a separable and complete metric. Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood. Miscellaneous Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense. Door space. A topological space is a door space if every subset is open or closed (or both). Topological Homogeneity. A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism such that Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous. Finitely generated or Alexandrov. A space X is Alexandrov if arbitrary intersections of open sets in X are open, or