TITLE: Applications of zero sets in topology
QUESTION [0 upvotes]: We call a subset $A$ of a topological space $X$ a zero set if it is of the form $$f^{-1}(0)\equiv\{x\in X:f(x)=0\}$$ for some continuous map $f:X\to\mathbb{R}$.
What applications do zero sets have in topology?
I know that my question seems more general but it would be good if someone can refer me to some references, like books or a thesis or a research paper. Thanks for any help.

REPLY [4 votes]: Look at the book "Rings of continuous functions" (Gilman and Jerrison) to see why they're useful.. Filters of zerosets correspond to ideals of the ring $C(X)$.
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