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

Hello;

If $\displaystyle X$ is a space and $\displaystyle H(X)$ denotes the group of homeomorphisms of X with composite between functions as a binary operation. My question is:

Show that the weight of $\displaystyle H(x)$ does not exceed the weight of $\displaystyle X$.

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 $\displaystyle W(X)$ is the smallest cardinality of a base of $\displaystyle X$. The Lindelof number $\displaystyle L(x)$ is the least infinite cardinal number http://eom.springer.de/c/images/c020.../c02036011.png such that every open covering of $\displaystyle X$ 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