  In a recent paper \cite{T} the fact that a class of locally compact metric
spaces $X$, among which are Euclidean spaces, are not homemorphic to their
punctured version $X\men\{p\}$, was given an interesting new proof which does
not use algebraic topology; essential tools of this proof are a boundedly
compact metric structure, and path--connectedness near infinity. Here we show
that local compactness and ordinary connectedness near infinity suffice; no
metrizability is needed, and moreover we can also delete whole compact subsets,
not only single points. Some non--homeomorphism results on many--holed
Euclidean balls are also obtained. This note ought to distil the essence of the
method developed in \cite{T}.
