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**Cardinality of the Preimage f^{-1}(y) of f:X-->Y continuous?**

Hi, All:

Let X,Y be topological spaces and f:X-->Y non-constant continuous function.

I'm curious as to whether it is possible for the fiber {f^{-1}(y)} of some y in Y

to be uncountable, given that the fiber is discrete (this condition is intended to

discount cases like f:ℝ-->ℝ, that are 0 in some interval, using, e.g., combinations

of e

^{-1/x2}) . There are examples of fibers being countably-

infinite, like in f:ℝ→ ℝ, f(x)=sinx, cosx, etc. , or the topologists sine curve on [0,1].

I suspect it may be possible, if X is metric to use f:X-->X with f(x)=d(x,S) , i.e.,

S is a subset of X , and d(x,S):=inf{d(x,s): s in S}. I thought S=Cantor set may work, but

the points of C are not isolated in ℝ (in [0,1], actually). Maybe if one can define an uncountable subset of

a metric space , all of whose points are isolated, we would be done.

Any IDeas?

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