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I'm writing an article on topology recently. I'm not good at English writing. It may be good for me to post this question somewhere else; however, I prefer to post here, as it is related to math.

My idea is this:

It is known that the cardinality of the compact metrizable space is always at most $2^\omega$. This is a necessary condition for a compact metrizable space. Therefore, it is necessary for one to discuss the cardinality of the space before trying to prove a space is compact metrizable.

Is this OK? Or is there a better way of expressing the idea? Thanks for your help.

(Sorry; I don't know how to tag it. I just want to attract one's attention.)

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    $\begingroup$ I'm a bit confused. I don't need to discuss the cardinality of the closed unit interval to say conclude that it is compact metrizable. I simply note that it is a compact subset of a metric space. Later I might note that it does indeed have cardinality continuum, but that was not important in my analysis. Perhaps it would be best to indicate how you plan to use the paragraph in question, and why you feel being so explicit is important. $\endgroup$ – user642796 Apr 9 '13 at 8:56
  • $\begingroup$ @ArthurFischer: if we can not decide a space is compact metrizable, then we may need to discuss the cardinality of the space. $\endgroup$ – Paul Apr 9 '13 at 9:04
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I can understand what you're saying, so at least that's OK. However, it could be shortened into one sentence, since they all seem to say the same.

Maybe the following shortening is clearer?

"A necessary condition for a space to be compact metrizable is that the cardinality is at most $ 2^\omega$. Therefore, we discuss/show the following:..."

The sentence starting with "Therefore" could be changed accordingly to what (or what not) you're going to do afterwards: show that space cannot be compact metrizable because of cardinality, or show that at least the space could be compact metrizable, considering the cardinality (which totally depends on the context whether or not this is interesting, I haven't seen this in practice, but I'm not a topologist).

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