Though there is a theory for von Neumann algebras on non-separable Hilbert spaces ( and indeed much of the general theory still holds in that case ) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces.
12.
If " H " 1 is a separable space ( in particular, if it is a Euclidean space ) the result is true in Zermelo Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.
13.
Marczewski proved that the topological dimension, for arbitrary metrisable separable space " X ", coincides with the Hausdorff dimension under one of the metrics in " X " which induce the given topology of " X " ( while otherwise the Hausdorff dimension is always greater or equal to the topological dimension ).
14.
If the requirement for the countable chain condition is replaced with the requirement that " R " contains a countable dense subset ( i . e ., " R " is a separable space ) then the answer is indeed yes : any such set " R " is necessarily order-isomorphic to "'R "'( proved by Cantor ).
15.
:: : I am going to stop now in case I get too carried away; ) but these problems illustrate the diversity of general topology and how simple ideas such as the OP's question may lead to such interesting questions ( for instance, in question one, we know that every separable space satisfies the countable chain condition; the obvious question is whether there exists a non-separable space which satisfies the countable chain condition ( that is, does the converse hold ? ).
16.
:: : I am going to stop now in case I get too carried away; ) but these problems illustrate the diversity of general topology and how simple ideas such as the OP's question may lead to such interesting questions ( for instance, in question one, we know that every separable space satisfies the countable chain condition; the obvious question is whether there exists a non-separable space which satisfies the countable chain condition ( that is, does the converse hold ? ).
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