Talk:Cosmic space

Disputed - take 1

This article contains errors. Even the definition ("the continuous image of some separable space") is wrong, as can be seen by checking the referenced book (or just by noting that it would be a ridiculous definition, as separability is preserved by continuous images). --Zundark (talk) 10:54, 21 October 2008 (UTC)

Disputed

The above-mentioned error in the definition has been corrected, but there is still a problem: the two "equivalent" properties are not equivalent. It is true that any continuous image of a separable metric space has a countable network, but the converse is false (any set of cardinality greater than c endowed with the trivial topology is a counterexample). Note that the referenced book is correct: it states the equivalence in the context of regular T₁ spaces, not for arbitrary topological spaces. --Zundark (talk) 10:40, 5 November 2008 (UTC)

If by c you mean the cardinality of the continuum (don't say later that this is not what you meant because this is well-accepted notation and if you had meant something else, you would have used a capital letter for an index set), then you are wrong. For instance, take the space of all continuous funtions from the unit interval to itself with the uniform metric. This is a separable metric space. Take the same set ([0,1]^[0,1]) but this time with the indiscrete topology. The identity morphism in the category of all sets from [0,1]^[0,1] to itself is also continuous as a function from a function space to an indiscrete space having cardinality greater than c.

Topology Expert (talk) 11:18, 5 November 2008 (UTC)

I did indeed mean the cardinality of the continuum, and I am right. The set of all continuous functions from the unit interval to itself has the cardinality of the continuum ("continuous" is important here!), so your example doesn't fit my prescription. In fact, every separable metric space has at most the cardinality of the continuum, which was my point. --Zundark (talk) 12:06, 5 November 2008 (UTC)

How about R^ω (the product of R (standard topology) with itself countably many times)? This space is the countable product of seperable metric spaces and therefore separable metric (has cardinality greater than c contrary to what you said about separable metric spaces on the last line of your post). The trivial topology on the same set (which has cardinality greater than c), is an example of an indiscrete cosmic space having cardinality greater than c.

Probably, what you intended to say was that any separable metric space has cardinality at most ${\beth _{2}}$ (in fact this is true for separable Hausdorff spaces).

Topology Expert (talk) 02:04, 6 November 2008 (UTC)

Well, your remark about separable Hausdorff spaces will do for the purpose of showing that sufficiently large sets with the trivial topology are counterexamples to the equivalence claimed in the article.

But for reference, I meant what I said about separable metric spaces. In particular, R^ω does have the cardinality of the continuum (link provided because that article lists books where proofs can be found). In detail:

$|\mathbb {R} ^{\omega }|=|\mathbb {R} |^{|\omega |}={\mathfrak {c}}^{\aleph _{0}}=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\aleph _{0}}=2^{\aleph _{0}}={\mathfrak {c}}.$

In general, since a separable metric space has a countable base, it has at most continuum many open sets, and since the complement of each singleton gives a different open set, there can't be more than continuum many points. --Zundark (talk) 09:31, 6 November 2008 (UTC)

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