in topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. in every topological space the empty set and the one-point sets are connected; in a totally disconnected space these are the only connected subsets. an important example of a totally disconnected space is the cantor set. another example, playing a key role in algebraic number theory, is the field q'p of p-adic numbers. ==definition== a topological space x is totally disconnected if the connected components in x are the one-point sets. ==examples== the following are examples of totally disconnected spaces: *discrete spaces * the rational numbers * the irrational numbers * the p-adic numbers; more generally, profinite groups are totally disconnected. * the cantor set * the baire space * the sorgenfrey line * zero-dimensional t1 spaces * extremally disconnected hausdorff spaces * stone spaces * the knasterâkuratowski fan provides an example of a connected space, such that the removal of a single point produces a totally disconnected space. * the erdÅs space âp(z)â©\mathbb{q}^{\omega} is a totally disconnected space that does not have  dimension zero. ==properties== *subspaces, products, and coproducts of totally disconnected spaces are totally disconnected. *totally disconnected spaces are t1 spaces, since singletons are closed. *continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the cantor set. *a locally compact hausdorff space is zero-dimensional if and only if it is totally disconnected. *every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces. *it is in general not true that every open set is also closed. *it is in general not true that the closure of every open set is open, i.e. not every totally disconnected hausdorff space is extremally disconnected. ==constructing a disconnected space== let x be an arbitrary topological space. let x\sim y if and only if y\in \mathrm{conn}(x) (where \mathrm{conn}(x) denotes the largest connected subset containing x). this is obviously an equivalence relation. endow x/{\sim} with the quotient topology, i.e. the coarsest topology making the map m:x\mapsto \mathrm{conn}(x) continuous. with a little bit of effort we can see that x/{\sim
General_topology Properties_of_topological_spaces