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Re: Connected Subspaces
Posted:
Jun 30, 2014 10:27 AM


In article <290620141155554340%edgar@math.ohiostate.edu.invalid>, Peter Flor <peter.flor@chello.at> wrote:
> The answer is contained essentially in the topology textbook by K. > Kuratowski. > Let M be a space as required: metric, connected, card(M)>1. Then M is > at least of continuum cardinality: > if 0<t<d(a,b) for some a and b in M then the set of all x such that > d(a,x) = t cannot be empty since M is connected. (If empty it would > furnish an obvious decomposition of M.) Thus we have continuum many > nonempty, pairwise disjoint sets in M, and a fortiori, continuum many > points in M.
Cute proof. Another proof. Since M is completely Hausdorff, for distince a,b in M, some Urysohn function f:M > [0,1] with f(a) = 0, f(b) = 1. Since M is connected, f(M) = [0,1]. Thus c <= M.
> Next, let M_1 be the set of those x{\in}M disconnecting M, and M_2 its > complement. Then at least one of M_1 and M_2 has the same > cardinality as M. All sets M\{x}, x{\in}M_2 , are connected, and they > are pairwise distinct. So if card(M_2) = card(M) we are done. Moreover > by a certain lemma (Thm. 4 in Kuratowski, Chapter 46 II), if x{\in}M_1 > and A{\cup}B is a decomposition of M\{x}, then both A{\cup}{x} and > B{\cup}{x} are connected. Obviously, couples [A{\cup}{x}, B{\cup}{x}] > are different for different values of x. Hence, the set of these > couples has the same cardinality as M_1. In either case, the set of > connected subsets of M has uncountable cardinality.
Additionally A \/ {x} or B \/ {x} = M.
Can you show for each point p, that p in in many M connected subsets. How many of those subsets would have cardinality M?



