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## Homework Statement

I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity.

## The Attempt at a Solution

So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is closed.

The set A can be written as A = {(x, 1/x) : x is in R\{0}}. Since f(x) = 1/x is a continuous function on R\{0}, and since any subset of R containing only one element is closed, f^-1{a} = {1/a} is a closed subset of R, for every a in R\{0}. Any ordered pair of the form (a, 1/a) can be written as a cartesian product of the sets {a} x {1/a}, which is closed, since the sets are closed. But an infinite union of such sets need not be closed. I feel it's warm around here, but I just can't figure it out.

Perhaps I'm not on the right track. Any help appreciated, as always.