topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A non-Hausdorff topological space is a topological space which is not a Hausdorff topological space.
Since the full subcategory of Hausdorff topological spaces inside the category of topological spaces is reflective, all limits in Top of diagrams of Hausdorff spaces are again Hausdorff spaces, but for colimits this fails in general (“non-Hausdorff quotients”, the colimit in the subcategory is given by applying Hausdorff reflection to the colimit formed in Top).
Hence examples of non-Hausdorff spaces generically arise from forming quotient topological spaces of Hausdorff spaces:
Consider the real line $\mathbb{R}$ regarded as the 1-dimensional Euclidean space with its metric topology and consider the equivalence relation $\sim$ on $\mathbb{R}$ which identifies two real numbers if they differ by a rational number:
Then the quotient topological space
is a codiscrete topological space, hence in particular a not a Hausdorff space.
Last revised on April 13, 2017 at 05:16:11. See the history of this page for a list of all contributions to it.