Connectedness of Subspaces

A subset $ A $ of a topological space $ X $ is said to be connected in $ X $ when, equipped with the subspace topology inherited from $ X $, the set $ A $ forms a connected topological space.

This perspective extends the notion of connectedness to any subset of a topological space, not only to the space as a whole.

In practice, the question is whether a particular subset remains connected once it is viewed with the topology induced by $ X $.

Note. Concretely, we take the set $ A $, endow it with the subspace topology inherited from $ X $, and ask whether $ A $ can be written as the union of two nonempty, disjoint sets that are open in this topology. If such a decomposition exists then $ A $ is disconnected in $ X $. If no such decomposition exists then $ A $ is connected in $ X $.

A practical example

Consider the real line $ \mathbb{R} $ with its standard topology, and the subset $ A $.

$$ A = [-1,0) \cup (0,1] $$

This set omits exactly one point, namely $ 0 $.

It contains all real numbers from -1 up to 0, excluding 0, and all real numbers from 0 to 1, again excluding 0.

The absence of that single point separates $ A $ into two distinct parts:

the interval from -1 to 0 (excluded)
the interval from 0 (excluded) to 1

These parts are:

$$ U = [-1,0) $$

$$ V = (0,1] $$

Both $ U $ and $ V $ are open in $ A $ when considered with the subspace topology.

They are disjoint, they do not intersect, and together they exhaust the entire set $ A $. This is precisely the standard characterization of disconnectedness.

$$ U \cap V = \emptyset $$

$$ U \cup V = A $$

Hence the subspace $ A $ is disconnected in $ \mathbb{R} $.