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
What is called the line with two origins is the topological space which results by “gluing” a copy of the real line identically to itself, except at the origin. This is a basic example of a non-Hausdorff topological space (even a non-Hausdorff smooth manifold) and of the fact that quotient topological spaces of Hausdorff topological spaces need not be Hausdorff themselves.
Consider the disjoint union $\mathbb{R} \sqcup \mathbb{R}$ of two copies of the real line $\mathbb{R}$ regarded with its metric topology. Moreover, consider the equivalence relation on the underlying set which identifies every point $x_i$ in the $i$th copy of $\mathbb{R}$ ($i \in \{0,1\}$) with the corresponding point in the other, the $(1-i)$th copy, except when $x = 0$:
The quotient topological space
by this equivalence relation is called the line with two origins.
The line with two origins is a non-Hausdorff topological space:
Because by definition of the quotient space topology, the open neighbourhoods of $0_i \in \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim$ are precisely those that contain subsets of the form
But this means that the “two origins” $0_0$ and $0_1$ may not be separated by neighbourhoods, since the intersection of $(-\epsilon, \epsilon)_0$ with $(-\epsilon, \epsilon)_1$ is always non-empty:
of the line with two origins is the real line itself:
It is also a topological manifold, even a smooth manifold, except that one often requires such to be Hausdorff to rule out examples such as this. In a similar vein, it is locally compact (and a compact version can be made by starting with $[{-1,1}]$ instead of $\mathbb{R}$), paracompact, and so forth.