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 connected topological space is called nilpotent if
its fundamental group is a nilpotent group;
the action of on the higher homotopy groups is nilpotent in that the sequence , terminates.
The central statement of rational homotopy theory says that the rational homotopy type of nilpotent topological spaces of finite type is equivalently reflected in nilpotent L-infinity algebras of finite type.
The rational homotopy theory of nilpotent topological spaces is discussed in
Aldridge Bousfield, V. K. A. M. Gugenheim, section 9.1 of On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)
Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. Volume 74, Number 2 (1978), 429-460. (euclid)
Last revised on March 1, 2017 at 05:45:08. See the history of this page for a list of all contributions to it.