topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, the Tychonoff plank is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

defined using

ordinal space In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...

s that is a

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...

to several plausible-sounding

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

s. It is defined as the

topological product In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...

of the two

,\omega_1 /math> and,\omega /math>, where \omega is the

first infinite ordinal In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...

and

\omega_1

the

first uncountable ordinal In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. Whe ...

. The deleted Tychonoff plank is obtained by deleting the point

\infty = (\omega_1,\omega)

Properties

The Tychonoff plank is a

compact Hausdorff space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

and is therefore a

normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...

. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not

completely normal In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...

. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not

perfectly normal ''Perfectly Normal'' is a Canadian comedy film directed by Yves Simoneau, which premiered at the 1990 Festival of Festivals, before going into general theatrical release in 1991. Simoneau's first English-language film, it was written by Eugene Lip ...

because it is not a G_δ space: the singleton

\

is closed but not a G_δ set. The

Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...

of the deleted Tychonoff plank is the Tychonoff plank.

Notes

References

* * *

External links

*{{MathWorld, urlname=TychonoffPlank, title=Tychonoff Plank, author=Barile, Margherita Topological spaces

Properties

Notes

See also

References

External links