Do each of the 1,110w:sig. ror, th, Theorem• 000 11 the resulting t…



Do each of the 1,110w:sig.
ror, th, Theorem•
000 11 the resulting topology What topology results when 13'= 0? when B = X? , Let B be it agricultural subset of x and for each nonempty A C X let A = A u B, wit. = 0. Describe the known
Lot -V be a marl.. extension whose topology is generated by a metric p. Pomp that tile metric d (kilned by (a.r.y) = 2. p(x,y) generates the similar topology on X. If in a flatten, all nearest fines are taken as subbasis, what is the topology? fl-Let (Tal a E Al be any fansily of topologies on X. Define U to be the topology having U0 T0 as subbasis. Prove that U is the lowest of the topologies on X larger that whole T0. .rove that bdy(A) = 0 if and merely if A is botls known and close. 8. Let D be consolidated in X. Prove that ThTU= U for whole known U ç x. 9. An known set U is denominated established if U = int(U); a close set A is denominated established 11 A = int(A). Prove that (a) If A is close, then int(A) is a established known set. (b) If U is known, then U is a established close set. 10. Let f X Y and g : Y Z. Prove that if f, g are known (closed), so too is g o f. 1"/".- Let X = U X2 and f : X Y. Assume flx, and fix, tsbecouctisiosousatxpx1nfl0. pomp that f is regular at a.

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