see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
Basic homotopy theory
The circle is a fantastic thing with lots and lots of properties and extra structures. It is a:
and it is one of the basic building blocks for lots of areas of mathematics, including:
We consider the circle first as a topological space, then as the homotopy type represented by that space.
The two most common definitions of the circle are:
It is the subspace of $\mathbb{C}$ consisting of those numbers of length $1$:
(of course, $\mathbb{C}$ can be identified with $\mathbb{R}^2$ and an equivalent formulation of this definition given; also, to emphasise the reason for the notation $U(1)$, $\mathbb{C}$ can be replaced by $\mathbb{C}^* = GL_1(\mathbb{C})$)
It is the quotient of $\mathbb{R}$ by the integers:
The standard equivalence of the two definitions is given by the map $\mathbb{R} \to \mathbb{C}$, $t \mapsto e^{2 \pi i t}$.
As a homotopy type the circle is for instance the homotopy pushout
In homotopy type theory, this can be formalized as a higher inductive type generated by a point base
and a path from base
to itself; see the references.
The topological circle is a compact, connected topological space. It is a $1$-dimensional smooth manifold (indeed, it is the only $1$-dimensional compact, connected smooth manifold). It is not simply connected.
The circle is a model for the classifying space for the abelian group $\mathbb{Z}$, the integers. Equivalently, the circle is the Eilenberg-Mac Lane space $K({\mathbb{Z}},1)$. Explicitly, the first homotopy group $\pi_1(S^1)$ is the integers $\mathbb{Z}$. But the higher homotopy groups $\pi_n(S^1) \simeq *$, $n \gt 1$ all vanish (and so is a homotopy 1-type). This can be deduced from the result that the loop space $\Omega S^1$ of the circle is the group ${\mathbb{Z}}$ of integers and that $S^1$ is path-connected. A proof of this in homotopy type theory is in Shulman P1S1.
The result that $S^1\simeq K({\mathbb{Z}},1)$ holds in a general Grothendieck (∞,1)-topos. In fact, more generally, for $X$ a pointed object of a Grothendieck (∞,1)-topos ${\mathcal{H}}$, there is a natural equivalence between the suspension object $\Sigma X$ and the classifying space $B{\mathbb{Z}}\wedge X$. In particular, when $X$ is specifically the 0-truncated two-point space $S^0$, we get $S^1\simeq K({\mathbb{Z}},1)$.
The product of the circle with itself is the ($2$)-torus
Generally, the $n$-torus $T^n$ is $(S^1)^n$.
A formalization in homotopy type theory, along with a proof that $\Omega S^1\simeq {\mathbb{Z}}$ (and hence $\pi_1(S^1) \simeq \mathbb{Z}$), can be found in
The proof is formalized therein using the Agda proof assistant. See also
The HoTT Book, section 8.1
The HoTT Coq library: theories/hit/Circle.v
The HoTT Agda library: homotopy/LoopSpaceCircle.agda