TITLE: CW-pairs are good pairs
QUESTION [5 upvotes]: Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?

REPLY [2 votes]: This should have been a comment to Stefan Hamcke's answer.
The homotopy in Stefan's answer seems to be identical to the one given in Hatcher's appendix.  Several details are left to the reader, though.  In particular, Hatcher does not explain why his homotopy is continuous.
Stefan Hamcke suggests deducing this from universal properties, but the topology on $I \times N_\epsilon(A)$ is obtained by combining three constructions: product topology, subspace topology, and quotient topology.  These don't always commute (and shouldn't be expected to, since it's commuting a limit and a colimit!)
Universal properties gives continuity when $I \times N_\epsilon(A)$ is given the quotient topology of the subspace topology, whereas what we need is continuity in the subspace topology of the quotient topology.