Baire property; first category; second category
Steinhaus  prove that if a set $A$ has a positive Lebesgue measure in the line then its distance set contains an interval. He obtained even stronger forms of this result in , which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar . In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus  and their corresponding form for ratios by Bose-Majumdar .
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