mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly

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 ...

, a cosmic space is any

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 ...

that is a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

of some separable

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

. Equivalently (for regular ''T''₁ spaces but not in general), a space is cosmic if and only if it has a

countable In mathematics, a set is countable if either it is 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 from it into the natural numbers; ...

network; namely a

collection of

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of the space such that any

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

is the

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

of a subcollection of these sets. Cosmic spaces have several interesting properties. There are a number of unsolved problems about them.

Examples and properties

* Any open subset of a cosmic space is cosmic since open subsets of separable spaces are separable. * Separable metric spaces are trivially cosmic.

Unsolved problems

It is unknown as to whether ''X'' is cosmic if: a) ''X''² contains no uncountable

discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

; b) the countable product of ''X'' with itself is hereditarily separable and hereditarily Lindelöf.

References

* * {{cite book , title=Encyclopedia of General Topology , url=https://archive.org/details/encyclopediagene00hart , url-access=limited , first1=K.P. , last1=Hart , first2=Jun-iti , last2=Nagata , first3=J.E. , last3=Vaughan , publisher=

Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', th ...

, year=2003 , isbn=0080530869 , pag
273

External links

A book of unsolved problems in topology; see page 91 for cosmic spaces
Topology Topological spaces