Topology is Locally Compact iff Ordered Set of Topology is Continuous

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $L = \left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$

Then
 * $(1): \quad T$ is locally compact implies $L$ is continuous
 * $(2): \quad T$ is regular space and $L$ is continuous implies $T$ is locally compact