Connected subset

A subset $A$ of a topological space $X$ is a connected subset when $A$, with the subspace topology inherited from $X$, is a connected space. Thus connectedness is tested inside $A$: a forbidden separation of a topological space consists of nonempty subsetswhere $O_1,O_2$ are open sets in $X$. The ambient open sets need not be disjoint outside $A$.

For example, in the real line with its usual topology, $[0,2]$ is a connected subset, whereas $\{0,2\}$ is not: its two singleton sets are open in its subspace topology. A subset of a connected space need not be connected. Conversely, $[-2,-1]$ is a connected subset of the disconnected topological space $[-2,-1]\cup [1,2]$. The ambient topological space need not itself be connected.

A finite set of distinct points in the real line inherits the discrete topology: a sufficiently small open interval around each point isolates that point. Its only nonempty connected subsets are singleton sets. This depends on the topology: a nonempty finite set with the indiscrete topology is a connected space, since its only open sets are itself and the empty set. Here, as in the definition of connected space, the empty set is also connected because it admits no separation of a topological space.^[ref]

The distinction matters when finding connected components: we search among connected subsets of the specified topological space, rather than judging whether the surrounding drawing looks like one piece.

Suppose connected subsets $U_j$ all meet one connected member $U_1$. A separation of their union would place $U_1$ entirely on one side. Each $U_j$ is also entirely on one side and intersects $U_1$, so every $U_j$ lies on the same side, contradicting nonempty separation. Thus the union is connected. If the subsets are path-connected, join two points by a path to $U_1$, a path inside $U_1$, and a path back out. This needs pairwise contact with the hub, not one point shared by every subset.

In a plane set with $x,y>0$ and rational ratio $x/y$, any irrational number $a$ lying between two attained ratios separates the set into the nonempty relatively open sets $x/y<a$ and $x/y>a$. This exhibits a failure of connectedness using the continuous slope map, even though each individual ray is path-connected.