View source for Totally-disconnected space

A [[topological space]] in which any subset containing more than one point is disconnected (cf. [[connected space]]). An equivalent condition is that the [[connected component]] of any point in the space is that point itself. The [[topological product]] and the [[topological sum]] of totally-disconnected spaces, as well as any subspace of a totally-disconnected space, are totally disconnected. Any totally-disconnected [[compactum]] is a [[zero-dimensional space]] (in every sense). Such compacta are important, in particular, because they are [[Stone space]]s of [[Boolean algebra]]s. A totally-disconnected space (a Knaster–Kuratowski fan) in the plane that can be made into a connected space as a result of the addition of a single point, has been constructed. Such a space is not zero-dimensional. In a [[Hilbert space]], the subspace formed by the points all coordinates of which are rational is totally disconnected and one-dimensional. If each point in a space is the intersection of all closed-and-open sets including it, the space is totally disconnected (in particular, all zero-dimensional spaces are totally disconnected). However, there exists a totally-disconnected [[metric space]] with a [[countable base]] in which not all points are intersections of such closed-and-open sets.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  W. Hurevicz,   G. Wallman,   "Dimension theory" , Princeton Univ. Press  (1948)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R. Engelking,   "Outline of general topology" , North-Holland  (1968)  (Translated from Polish)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Springer  (1989)  (Translated from French)</TD></TR></table>



====Comments====
There is a totally-disconnected plane set $E$, no proper superset of which is totally disconnected, [[#References|[a3]]]. The complement of such a set is called a primitive dispersion set for the plane. For all $n$, there are $n$-dimensional totally-disconnected separable metric groups, [[#References|[a4]]].

There is some confusion in the terminology concerning disconnected spaces. There are a few kinds of disconnectedness; the two most common notions are: i) the one in the article: connected subsets consist of at most one point; and ii) for every two points $x$ and $y$ there is a closed-and-open set $C$ such that $x\in C$ and $y\notin C$.

Both are called total disconnectedness at times. References [[#References|[a1]]] and [[#References|[a2]]] call spaces satisfying ii) totally disconnected, and in [[#References|[a2]]] spaces satisfying i) are called hereditarily disconnected (because they have no non-trivial connected subspaces). (Note that ii) implies i).)

The Knaster–Kuratowski fan is a subset of the plane defined as follows: Consider the usual Cantor middle-third set $C$ situated in the interval $[0,1]\times\{0\}$ in the plane. Connect every point $x$ of $C$ with the point $(1/2,1/2)$ by the line segment $L_x$. For each $x\in C$ take a subset $F_x$ of $L_x$ as follows: if $x$ is an end-point of an interval in the complement of $C$ take all points of $L_x$ with rational second coordinate, otherwise take the points with irrational second coordinate. The union $\bigcup_{x\in C}F_x$ is the Knaster–Kuratowksi fan. When one removes the point $(1/2,1/2)$ from $F$, one obtains a space that satisfies i) above but not ii). See also [[Kuratowski–Knaster fan|Kuratowski–Knaster fan]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M.E. Estill,   "A primitive dispersion set of the plane"  ''Duke Math. J.'' , '''9''' :  19  (1952)  (323–328)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Mill,   "$n$-dimensional totally disconnected topological groups"  ''Math. Japon.'' , '''32'''  (1987)  pp. 267–273</TD></TR>
</table>