In
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 ...
, particularly
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a cosmic space is any
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of some
separable 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 ...
. Equivalently (for
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
''T''1 spaces but not in general), a space is cosmic if and only if it has a
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 ...
network; namely a
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 ...
collection of
subset
In mathematics, a 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 a ...
s of the space such that any
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
is the
union 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 (journal), Cell'', the ScienceDirect collection of electronic journals, ...
, year=2003 , isbn=0080530869 , pag
273
External links
A book of unsolved problems in topology; see page 91 for cosmic spaces
Topology
Topological spaces