Totally disconnected

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.

A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X are the one-point sets.

The following are examples of totally disconnected spaces:

Let ${\displaystyle X}$ be an arbitrary topological space. Let ${\displaystyle x\sim y}$ if and only if ${\displaystyle y\in \mathrm {conn} (x)}$ (where ${\displaystyle \mathrm {conn} (x)}$ denotes the largest connected subset containing ${\displaystyle x}$). This is obviously an equivalence relation whose equivalence classes are the connected components of ${\displaystyle X}$. Endow ${\displaystyle X/{\sim }}$ with the quotient topology, i.e. the finest topology making the map ${\displaystyle m:x\mapsto \mathrm {conn} (x)}$ continuous. With a little bit of effort we can see that ${\displaystyle X/{\sim }}$ is totally disconnected. We also have the following universal property: if ${\displaystyle f:X\rightarrow Y}$ a continuous map to a totally disconnected space ${\displaystyle Y}$, then there exists a unique continuous map ${\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}$ with ${\displaystyle f={\breve {f}}\circ m}$.

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