Tychonoff in a sentence

More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K. In fact, one can always choose K to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals ).

Some authors add the assumption that the starting space X be Tychonoff (or even locally compact Hausdorff), for the following reasons: *The map from X to its image in βX is a homeomorphism if and only if X is Tychonoff.

Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff).

An example of a regular space that is not completely regular is the Tychonoff corkscrew.

Every locally compact Hausdorff space is Tychonoff.

Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff).

Examples and counterexamples Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular.

If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX.

In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.

In Wikipedia, we will use the terms "completely regular" and "Tychonoff" freely, but we'll avoid the less clear "T" terms.

It follows that: * Every subspace of a completely regular or Tychonoff space has the same property.

It follows that if a topological group is T 0 ( Kolmogorov ) then it is already T 2 ( Hausdorff ), even T 3½ ( Tychonoff ).

On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.

Other examples include: * Every metric space is Tychonoff; every pseudometric space is completely regular.

Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces.

Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies.

The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice.

The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness ( Tychonoff's theorem ) and the T 2 axiom are preserved under arbitrary products.

Tychonoff) if and only if each factor space is completely regular (resp.

Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact.