mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Sierpiński's theorem is an isomorphism theorem concerning certain

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s, named after

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...

who proved it in 1920. It states that any

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

without

isolated point In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton i ...

s is

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

\mathbb

(with its standard topology).

Examples

As a consequence of the theorem, the metric space

\mathbb^2

(with its usual

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...

) is homeomorphic to

\mathbb

, which may seem counterintuitive. This is in contrast to, e.g.,

\mathbb^2

, which is not homeomorphic to

\mathbb

. As another example,

\mathbb \cap

, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

/math> is also homeomorphic to

\mathbb

, again in contrast to the closed real interval

/math>, which is not homeomorphic to

\mathbb

(whereas the open interval

(0, 1)

is).

References

{{ reflist , refs = {{ cite journal , last = Sierpiński , first = Wacław , title = Sur une propriété topologique des ensembles dénombrables denses en soi , date = 1920 , journal =

Fundamenta Mathematicae ''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical sys ...

, volume = 1 , pages = 11–16 {{ cite journal , last = Błaszczyk , first = Aleksander , title = A Simple Proof of Sierpiński's Theorem , journal =

The American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposito ...

, volume = 126 , issue = 5 , pages = 464–466 , doi = 10.1080/00029890.2019.1577103 {{ cite web , last = Dasgupta , first = Abhijit , title = Countable metric spaces without isolated points , url = http://at.yorku.ca/p/a/c/a/25.pdf {{ cite book , last = Engelking , first = Ryszard , author-link = Ryszard Engelking , title = General Topology , publisher = Heldermann Verlag, Berlin , year = 1989 , isbn = 3-88538-006-4 , at = Exercise 6.2.A(d), p. 370 {{ cite book , last = Kechris , first = Alexander S. , title = Classical Descriptive Set Theory , year = 1995 , publisher = Springer , series =

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...

, at = Exercise 7.12, p. 40 {{ cite book , last = van Mill , first = Jan , title = The Infinite-Dimensional Topology of Function Spaces , year = 2001 , publisher = Elsevier , at = Theorem 1.9.6, p. 76 , isbn = 9780080929774

Examples

References

See also