# 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.

## Contents

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)

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.