Connected space

A topological space that cannot be represented as the sum of two parts separated from one another, or, more precisely, as the union of two non-empty disjoint open-closed subsets. A space is connected if and only if every continuous real-valued function on it takes all intermediate values. The continuous image of a connected space, the topological product of connected spaces, and the space of closed subsets of a connected space in the Vietoris topology are connected spaces. Every connected completely-regular space has cardinality not less than the cardinality of the continuum (if contains more than one point), although there also exist countable connected Hausdorff spaces.

Comments

For Vietoris topology see Hyperspace.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

A separation in a topological space is a non-trivial open-closed subset. A space is thus disconnected if and only if it has a separation, and totally separated if for any points $x \neq y$ there is a separation containing $x$ and not $y$.

The equivalence relation on a space $X$ that $x \equiv y$ if and only if for every separation $S$ of $X$ either $x$ and $y$ are in $S$ or $x$ and $y$ are in $X \setminus S$ defines equivalence classes called quasicomponents. The quasicomponent of $x$ is the intersection of all open-closed sets of $X$ containing $x$. If there is only one quasicomponent in $X$ then it is connected. A space is totally separated if and only if its quasicomponents are all singletons.

The equivalence relation on a space $X$ that $x \equiv y$ if and only if there is a connected subspace of $X$ containing $x$ and $y$ defines as classes the connected components: these are the maximal connected subspaces of $X$. The component of $x$ is the union of all connected subsets of $X$ containing $x$. A space is totally disconnected if and only if its connected components are all singletons.

See also: Extremally-disconnected space.

References

[b1]	Steen, Lynn Arthur; Seebach, J.Arthur jun. Counterexamples in topology (2nd ed.) Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001