For a given topological dynamical system
$T:X\rightarrow X$
over a compact set
$X$
with a metric
$d$
, the variational principle states that
$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=h_{d}(T),\end{eqnarray}$$
where
$h_{\unicode[STIX]{x1D707}}(T)$
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 Patrão [Entropy and its variational principle for non-compact metric spaces.
Ergod. Th. & Dynam. Sys.
30 (2010), 1529–1542], 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
$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=\min _{d}h_{d}(T),\end{eqnarray}$$
where the minimum is taken over every distance compatible with the topology of
$X$
. In the present work, we drop the properness assumption and extend the above result for any continuous map
$T$
. We also apply our results to extend some previous formulae for the topological entropy of continuous endomorphisms of connected Lie groups that were proved in Caldas and Patrão [Dynamics of endomorphisms of Lie groups.
Discrete Contin. Dyn. Syst.
33 (2013). 1351–1363]. In particular, we prove that any linear transformation
$T:V\rightarrow V$
over a finite-dimensional vector space
$V$
has null topological entropy.