see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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
Theorems
The ordinary “line” $\mathbb{R}$ (i.e. the real numbers equipped with their Euclidean metric topology) may be thought of as the result of gluing a countable set of copies of the half-open interval $[0, 1)$ end-to-end in both directions. A long line is similarly obtained by gluing an uncountable set of copies of $[0, 1)$ end-to-end in both directions, and is demonstrably “longer” than an ordinary line on account of a number of peculiar properties.
The long line is a source of many counterexamples in topology.
Let $\omega_1$ be the first uncountable ordinal, and consider the half-open interval $[0, 1)$ as an totally ordered set. A long ray is the ordered set $\omega_1 \times [0, 1)$ taken in the lexicographic order; as a space, it is given the order topology. The long line is the space obtained by gluing two long rays together at their initial points.
The long line is a line in the sense of being a $1$-dimensional locally Euclidean space (without boundary) that is not closed (so not a circle). However, it is not paracompact, so it is not homeomorphic to the real line (even though it is Hausdorff).
Let $L$ denote the long line, and $R$ the long ray.
Every continuous function $f\colon L \to \mathbb{R}$ is eventually constant, i.e., there exists $x \in L$ and $c \in \mathbb{R}$ such that $f(y) = c$ whenever $y \geq x$ (and similarly $f$ is constant for all sufficiently small $x$).
$L$ is a normal ($T_4$) space, but the Tychonoff product $L \times \bar{L}$ with its one-point compactification is not normal. (See for example Munkres.)
Every continuous map $L \to L$ has a fixed point.
$R$ and $L$ are sequentially compact but not compact. Thus, the image $f(L)$ under a continuous map $f: L \to \mathbb{R}$, being sequentially compact, is a compact subspace of $\mathbb{R}$.
The long line is not contractible. Proof sketch: Suppose $H \colon I \times L \to L$ is a homotopy such that $H(0, -)$ is constant and $H(1, -)$ is the identity. For each $t \in [0, 1]$ the image $\im H(t, -)$ is an interval (either bounded or unbounded), since $L$ is connected. One may show the set
is both closed and open. It also contains $0$, hence is all of $I$. But it can’t contain $t = 1$, contradiction.
Steen and Seebach, Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.
James Munkres, Topology (2nd edition). Prentice-Hall, 2000.
See also