nLab Tychonoff space

Idea

A Tychonoff space is a subspace of a compactum. By default, one usually means a Tychonoff topological space, although there exist (even classically) additional Tychonoff locales.

Definitions

The classical definition is:

Definition

A topological space $X$ is completely regular if, given a point $a$ of $X$ and a closed subset $F$ of $X$, if $a \notin F$, then there is a continuous map $f$ from $X$ to the real line $\mathbb{R}$ such that $f$ is constant when restricted to $F$ but takes a different value at $a$.

(Traditionally, one requires $f(a) = 0$ and $f(F) = \{1\}$, but this can always be forced by postcomposition with an affine linear map. One sometimes sees, even more specifically, the requirement that $f$ land in the unit interval $[0,1]$, but this too can be forced since $[0,1]$ is a retract of $\mathbb{R}$.)

An equivalent definition focussing on open sets is this:

Definition

A topological space $X$ is completely regular if, given a point $a$ of $X$ and an open neighbourhood $U$ of $a$ in $X$, there is a continuous $f\colon X \to \mathbb{R}$ and a real number $c$ such that $a \in f^*(\{c\}') \subseteq U$.

(Here $f^*(\{c\}')$ is the preimage under $f$ of the complement of the singleton of $c$. Again, one can force $c = 1$, $f(a) = 0$, and even $f\colon X \to [0,1]$ if desired.) This definition is suitable for constructive mathematics based on weak countable choice, Markov's principle, and the fan theorem (all of which follow from excluded middle), or in any case if the following interpretations are made:

• $f$ lands in the locale of real numbers,
• the complement $\{c\}'$ is the open subspace apart from $c$.

There is also a definition entirely in terms of the lattice of open sets, suitable for locales, which I need to look up again. (It's similar to the localic definition of regular space, but using a stronger notion than well-inside.)

When a space is completely regular, we also often ask it to be $T_0$:

Definition

(of $T_0$) Given any two points, if each neighbourhood of either is a neighbourhood of the other, then they are equal.

A completely regular $T_0$-space is variously called a $T_{3\frac{1}{2}}$-space (or $T_{\pi}$-space), a completely regular Hausdorff space (since it is a theorem that such a space is Hausdorff), or a Tychonoff space (or using other spellings of Tychonoff's name).

As is usual with the separation axioms, some authors while conflate the meanings of ‘completely regular’ and ‘$T_{3\frac{1}{2}}$’ either way, or even reverse them. In contrast, the terms ‘completely regular Hausdorff’ and ‘Tychonoff’ are unambiguous. The only unambiguous term for a completely regular space in general seems to be ‘$R_{2\frac{1}{2}}$’, but this is not widely recognised.

Properties

• Every completely regular space is preregular, so every completely regular Hausdorff space (as defined above) really is Hausdorff.

• Every completely regular space is regular.

• The natural map from a Tychonoff space to its Stone-Cech compactification (the unit of the adjunction) really is a compactification in that it is the inclusion of a dense subspace into a compact space. Conversely, any subspace of a compact Hausdorff space must be Tychonoff.

• Without $T_0$, we can say that a completely regular space embeds as a dense subspace of a compact regular space, and any subspace of a compact regular space is complete regular.

Examples

• Every metric space is Tychonoff (and every pseudometric space is completely regular).

• Every topological group is Tychonoff, if one requires groups to be Hausdorff; if not, then they are still completely regular.

• Every topological manifold is Tychonoff, if one requires manifolds to be Hausdorff. (But if not, then the non-Hausdorff manifolds are not completely regular, indeed not even preregular, and in fact they are still $T_0$, indeed $T_1$.)

• Every CW-complex is Tychonoff.

References

Named after A. N. Tychonoff.

For a version appropriate to constructive (but impredicative) mathematics, see

• Bernhard Banaschewski, Aleš Pultr; 2002; A Constructive View of Complete Regularity; web.