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March 21

Clopen, non-empty, proper subsets of R

Every subset of R is clopen with respect to the discrete topology. What is the coarsest topology on R for which there exists a clopen, non-empty, proper subset of R? — Fly by Night (talk) 20:10, 21 March 2011 (UTC)[reply]

The answer is nonunique, since the toplogies on R are not linearly ordered, even assuming that the question is what is the coarsest topology refining the usual topology on R. One coarsest topology that refines the usual one is obtained by taking the usual topology, and then in addition declaring as open any set of the form (-∞,0]∪U where U is open in the usual topology. Sławomir Biały (talk) 21:07, 21 March 2011 (UTC)[reply]

Thanks Sławomir, what's the cardinality of this topology? — Fly by Night (talk) 00:31, 22 March 2011 (UTC)[reply]

It has the cardinality of the continuum. Sławomir Biały (talk) 00:45, 22 March 2011 (UTC)[reply]

Could you explain why, please? The cardinality is the same as the set of all open subsets of R. The set of all subsets of R has cardinality 2^ℵ₁. — Fly by Night (talk) 00:57, 22 March 2011 (UTC)[reply]

Because the real line is a hereditarily Lindelöf space. (In particular, every open set is a countable union of intervals.) Sławomir Biały (talk) 02:06, 22 March 2011 (UTC)[reply]

Okay, a space X is Lindelöf if every open cover has a countable subcover. I very much appreciate your efforts on the maths reference desk, but you don't often leave enough explanation. I don't see how a relaxation of the compactness condition relates to the cardinality of a non-compact set. Why does the set of all open subsets have a different cardinality to the set of all subsets? Please give details, and not general theorems. — Fly by Night (talk) 02:35, 22 March 2011 (UTC)[reply]

Herediatrily Lindelof is probably not the best article to link to; this is something quite classical but I couldn't find the right article. An open set of the reals is a disjoint union of countable many open intervals, so the topology can be identified with the set of its endpoints. This sets up an injective map from the topology to the set of all countable families of pairs of real numbers. This has the cardinality of the continuum. Sławomir Biały (talk) 10:55, 22 March 2011 (UTC)[reply]

The coarsest example I can think of is where the only non-trivial open sets are a point x and R - {x}. More generally, replace x by any subset of R. AndrewWTaylor (talk) 21:16, 21 March 2011 (UTC)[reply]

Thanks Andrew, what's the cardinality of this topology? — Fly by Night (talk) 00:31, 22 March 2011 (UTC)[reply]

4. R, R - {x}, {x} and {}.--203.97.79.114 (talk) 20:54, 22 March 2011 (UTC)[reply]