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
The plane is the Cartesian space $\mathbb{R}^2$. This is naturally a topological space, a manifold, and a smooth manifold. If we take one of the axes (traditionally the second) to be imaginary, then this real plane may be identified with the complex plane $\mathbb{C}^1$. As a stage for Euclidean geometry, it may be called the Cartesian plane, Euclidean plane, or coordinate plane.
For discussion of the plane via axioms for the points and lines in it (synthetic geometry) see the references at Euclidean geometry.
Last revised on September 17, 2018 at 03:53:38. See the history of this page for a list of all contributions to it.