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.)
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).