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In the John's Kelley "General Topology" (pages 32-34) there is definition of both of them and its look like its just different ways of notation of one proposition. But also there is something strange on page 34 that is called "proof" and perhaps is just a sequence of random words.

Nicky Hekster
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Andrey Komisarov
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    It's okay to ask about proofs written in books, and about the difference between HMP and ZL, but this is not the way to do so. This is the way to alienate the very people from whom you've asked for help. Be respectful of the text, and be aware that most people will not have a copy of the book to look and clarify exactly what is going on, so you need to make your post self-contained to the best of your abilities. – Asaf Karagila Jun 25 '20 at 12:15
  • I removed an abbreviation you should not be using here. – Nicky Hekster Jun 25 '20 at 13:41
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    Lol at the tag "fake proofs". Any proof looks like a "sequence of random words" to those without the background or mathematical maturity to understand it. – Alex Kruckman Jun 25 '20 at 19:47

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It’s perfectly straightforward: the Hausdorff maximal principle is the special case of the Kuratowski lemma in which the partial order is some family of sets ordered by $\subseteq$. It just happens that the more general result follows from the special case — as is shown on page $34$. This should not be an unfamiliar phenomenon: in elementary calculus one learns that the Mean Value Theorem is easily proved from Rolle’s Theorem, which is clearly just a special case of it.

Brian M. Scott
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  • I know about proving general results by special, I seen the proof of Taylor series for $\mathbb{R}^n$ from one-dimension case. And is it true that all action is in proving that exist some bijection with respect to something? – Andrey Komisarov Jun 25 '20 at 20:27
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    @AndreyKomisarov: Absolutely not. – Brian M. Scott Jun 26 '20 at 04:57