Open Subset of Locally Connected Space is Locally Connected/Lemma

Lemma to Open Subset of Locally Connected Space is Locally Connected

Let $T = \struct {S, \tau}$ be a topological space.

Let $U$ be a connected set.

Let $A$ be a subset of $T$ such that $U \subseteq A$.

Let $\tau_A$ be the subspace topology on $A$ induced from $T$.

Then $U$ is connected in the topological subspace $\struct {A, \tau_A}$.

Proof

Let $\tau_U$ denote the subspace topology on $U$ induced from $T$.

By the definition of a connected set, the topological space $\struct {U, \tau_U}$ is connected.

Let ${\tau_U}'$ denote the subspace topology on $U$ induced from $\struct {A, \tau_A}$.

By Subspace of Subspace is Subspace, ${\tau_U}' = \tau_U$.

Thus the topological spaces $\struct {U, {\tau_U}'}$ and $\struct {U, \tau_U}$ are identical.

In particular, since $\struct {U, \tau_U}$ is connected, so is $\struct {U, {\tau_U}'}$.

Applying the definition of a connected set once more, it follows that $U$ is connected when considered as a subset of $A$.

$\blacksquare$