  For a given topological dynamical system $(X,T)$ over a compact set $X$ with
a metric $d$, the "variational principle" states that \begin{equation*}
\sup_{\mu}h_\mu(T) = h(T) = h_d(T), \end{equation*} where $h_\mu(T)$ is the
Kolmogorov-Sinai entropy, with the supremum taken over every $T$-invariant
probability measure, $h_d(T)$ is the Bowen entropy, and $h(T)$ is the
topological entropy as defined by Adler, Konheim and McAndrew. In [9], the
concept of topological entropy was adapted for the case where $T$ is a proper
map and $X$ is locally compact separable and metrizable, and the variational
principle was extended to \begin{equation*} \sup_{\mu}h_\mu(T) = h(T) = \min_d
h_d(T), \end{equation*} where the minimum is taken over every distance
compatible with the topology of $X$. In the present work, we dropped the
properness assumption, extending the above result for any continuous map $T$.
  We also apply our results to extend some previous formulas for the
topological entropy of continuous endomorphisms of connected Lie groups proved
in [4]. In particular, we prove that any linear transformation $T: V \to V$
over a finite dimensional vector space $V$ has null topological entropy.
