topology of infinite union of hyperplanes

Hi all:

I am working on Functional Analysis. I encounter a topology problem in my study of spectrum of certain operators, and it has bothered me for quite some time. Any idea or references is greatly appreciated.

Suppose M is an infinite (possibly uncountable) union of complex hyperplanes in $C^n$ . To be specific, we write $M=∪H_a$ where $H_a ={z∈C^n : acdot z=0}$ .

If M is a finite union, then the de Rham cohomology (with complex coefficient) of M c is generated by the 1-forms $acdot dz/ acdot z$ . This is a well-known theorem. My question is whether there is a similar theorem for an infinite union of hyperplanes. We can assume $M^c$ is open and nice, in particular we assume the first de Rham cohomology $H^1 (M^c , C)$ is finite dimensional. Then is $H^1 (M^c , C)$ spanned by the 1-forms $acdot dz/ acdot z$ ?

Thanks a lot!

Ron

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