Coarser Topology than Compact Space is Compact

Theorem
Let $S$ be a set.

Let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1$ is coarser than $\tau_2$:
 * $\tau_1 \subseteq \tau_2$

Let $\struct {S, \tau_2}$ be a compact space.

Then $\struct {S, \tau_1}$ is also compact.

Proof
Let $\struct {S, \tau_2}$ be a compact space as asserted.

Let $I_S: \struct {S, \tau_2} \to \struct {S, \tau_1}$ denote the identity mapping on $S$:
 * $\forall x \in S: \map {I_S} x = x$

From Identity Mapping to Coarser Topology is Continuous, $I_S$ is continuous.

We also have the result Identity Mapping is Surjection.

Hence we apply Compactness is Preserved under Continuous Surjection.

The result follows.