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The weight of H(X) does not exceed the weight of X?

Hello;

If is a space and denotes the group of homeomorphisms of X with composite between functions as a binary operation. My question is:

Show that the weight of does not exceed the weight of .

I found the proof in the attached file. But the problem is I could not understand it. So kindly read it and explain to me or if you have any other idea please help me.

The weight of a spaces is the smallest cardinality of a base of . The Lindelof number is the least infinite cardinal number http://eom.springer.de/c/images/c020.../c02036011.png such that every open covering of has a subcovering of cardinality http://eom.springer.de/c/images/c020.../c02036013.png.

Please help me and every guidance is highly appreciated.

Thaaaaaaaaaaank you in advance