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
Traditionally, a compactification of a topological space $X$ is a compact space $C$ together with an embedding $i \colon X \to C$ as a dense subspace. Well known is for instance the one-point compactification of a locally compact Hausdorff space which for instance sends the real line to the circle by adding a point at infinity.
In some cases, the terminology ‘compactification’ is applied to generalizations where $i$ is not a dense embedding (does not map homeomorphically to an image that is dense). For example, the one-point compactification of an already compact Hausdorff space adds an isolated point at infinity, and so is not a dense subspace. Or, if $X$ is not a Tychonoff space, then the universal map to its Stone-Cech compactification is not an embedding.
If the space has further geometric structure, the compactification is usually required to has such a structure and embedding has to preserve it. Many moduli spaces in algebraic and differential geometry have their natural compactifications. They are often useful because they carry natural integration which is useful in defining various invariants.
A useful intuition throughout is that a ‘compactification’ is a process of adding “ideal points at infinity” in some way to “complete” a space. (Compact regular spaces $X$ themselves being “complete” in a technical sense: there is a unique uniform structure whose uniform topology is the topology on $X$, and $X$ is complete with respect to this uniformity.)
Often a space can be viewed as a total space of a bundle over some base. We may want to embed the space into a bigger bundle, such that the induced embedding of each fiber into the new fiber is a compactification.
This is roughly the case in most compactifications in physics. In most cases the space is equipped with a Riemannian metric and additional quantities for defining physics, like a Lagrangian density, which possibly depend on the metric.
Then one requires that the compactified fiber is finite but small compared to some reference scale (or even viewed in a limit when the Riemannian volume tends to zero), see at Kaluza-Klein mechanism. Often one does not even consider a noncompact case to start with but by compactification in physics means only passing to the limit of small (Riemannian volume of) fibers.
A compactification of a topological space $X$ is a compact Hausdorff topological space $Y$ equipped with an embedding $X \hookrightarrow Y$ such that the closure of $X$ in $Y$ is the compact space: $\overline{X} = Y$.
An equivalence of two compactifications $Y_1$, $Y_2$ of $X$ is a homeomorphism $h \;\colon\; Y_1 \longrightarrow Y_2$ that preserves the inclusion of $X$.
In some sense the one-point compactification is the smallest possible compactification, while the Stone-Cech compactification is the largest. The following gives conditions that all notions of compactification agree.
For $X$ a Tychonoff space the following are equivalent:
There is a unique (up to equivalence) compactification, def. 1.
$X$ is already compact or its Stone-Cech compactification $\beta X$ adds a single point $\vert \beta X \backslash X\vert = 1$.
If two closed subsets of $X$ are completeley separated, then one of them is a compactum.
One place where this appears is (Hewitt 47).
The topological spaces satisfying the conditions of prop. 1 are also called almost compact topological spaces?.
wonderful compactification?