  We extend the definition of topological pressure to locally compact Hausdorff
spaces, and we demonstrate a "variational principle" comparing the topological
and measure theoretic pressures. Given a continuous $\mathbb{Z}_+^N$-action $T$
over a locally compact Hausdorff space $X$ and a continuous function vanishing
at infinity $f \in C_0(X)$, we define topological pressure $P(T,f)$ using open
covers of a special type we call "admissible covers". With this topological
pressure, we demonstrate that \begin{equation*}
  P(T,f)
  =
  \sup_{\mu}
  P_\mu(T,f), \end{equation*} where the supremum is taken over all
$T$-invariant probability Radon measures over $X$, and is equal to $0$ when
there is none.
  In the last section, we present an example that illustrates why admissible
covers are so adequate to deal with the non-compact case, while some other
approaches would fail.
