Explore: Compactifications De Hausdorff

one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact. Of particular interest are Hausdorff compactifications, i.e

compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space

one-point compactification. By the same construction, every locally compact Hausdorff space X is an open dense subspace of a compact Hausdorff space having

other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others

locally compact Hausdorff space that is not compact, then the one-point compactification of X {\displaystyle X} is a perfect, compact Hausdorff space. Therefore

Simply connected space Path connected space T0 space T1 space Hausdorff space Completely Hausdorff space Regular space Tychonoff space Normal space Urysohn's

topology Weak topology Compactifications include: Alexandroff extension Projectively extended real line Bohr compactification Eells–Kuiper manifold Projectively

compact group if the underlying topological space is locally compact and Hausdorff; a topological group is abelian if the underlying group is abelian. Examples

regular Hausdorff space embeds in a compact Hausdorff space (or, can be "compactified".) This construction is the Stone–Čech compactification. Conversely

dense subset is necessarily connected itself. Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous