18,889 research outputs found
Products of straight spaces
A metric space X is straight if for each finite cover of X by closed sets,
and for each real valued function f on X, if f is uniformly continuous on each
set of the cover, then f is uniformly continuous on the whole of X. A locally
connected space is straight if it is uniformly locally connected (ULC). It is
easily seen that ULC spaces are stable under finite products. On the other hand
the product of two straight spaces is not necessarily straight. We prove that
the product X x Y of two metric spaces is straight if and only if both X and Y
are straight and one of the following conditions holds: (a) both X and Y are
precompact; (b) both X and Y are locally connected; (c) one of the spaces is
both precompact and locally connected. In particular, when X satisfies (c), the
product X x Z is straight for every straight space Z. Finally, we characterize
when infinite products of metric spaces are ULC and we completely solve the
problem of straightness of infinite products of ULC spaces.Comment: 21 page
On locally 1-connectedness of quotient spaces and its applications to fundamental groups
Let be a locally 1-connected metric space and be
connected, locally path connected and compact pairwise disjoint subspaces of
. In this paper, we show that the quotient space
obtained from by collapsing each of the sets 's to a point, is also
locally 1-connected. Moreover, we prove that the induced continuous
homomorphism of quasitopological fundamental groups is surjective. Finally, we
give some applications to find out some properties of the fundamental group of
the quotient space .Comment: 11 page
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