  Co-compact entropy is introduced as an invariant of topological conjugation
for perfect mappings defined on any Hausdorff space(compactness and
metrizability not necessarily required). This is achieved through the
consideration of co-compact covers of the space. The advantages of co-compact
entropy include: 1) it does not require the space to be compact, and thus
generalizes Adler, Konheim and McAndrew's topological entropy of continuous
mappings on compact dynamical systems, and 2) it is an invariant of topological
conjugation, compared to Bowen's entropy that is metric-dependent. Other
properties of co-compact entropy are investigated, e.g., the co-compact entropy
of a subsystem does not exceed that of the whole system. For the linear system
(R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen's
entropy for this system is at least log 2. More general, it is found that
co-compact entropy is a lower bound of Bowen's entropies, and the proof of this
result generates the Lebesgue Covering Theorem to co-compact open covers of
non-compact metric spaces, too.
