In
mathematics, 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 po ...
is said to be ultraconnected if no two nonempty
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
s are
disjoint.
[PlanetMath] Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no
T1 space with more than one point is ultraconnected.
[Steen & Seebach, Sect. 4, pp. 29-30]
Properties
Every ultraconnected space
is
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
(but not necessarily
arc connected). If
and
are two points of
and
is a point in the intersection
, the function
defined by
if
,
and
if
, is a continuous path between
and
.
Every ultraconnected space is
normal,
limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
, and
pseudocompact
In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of p ...
.
[
]
Examples
The following are examples of ultraconnected topological spaces.
* A set with the indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
.
* The Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is na ...
.
* A set with the excluded point topology.
* The right order topology
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, t ...
on the real line.[Steen & Seebach, example #50, p. 74]
See also
* Hyperconnected space
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
Notes
References
*
* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) ...
''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition).
Properties of topological spaces