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 sets 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 (but not necessarily
arc 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 t ...
). 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
,
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 ps ...
.
[
]
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 name ...
.
* A set with the excluded point topology In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:T = \ \cup \
of subsets of ''X'' is then the excluded ...
.
* 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, th ...
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) h ...
''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition).
Properties of topological spaces