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Let x be any space. Define a relation p c on x as follows: x p c y if there is a path in x

Let x be any space. Define a relation pc on x as follows: xpcy if there is a path in x
from x to y.
(a) Show that pc is an equivalence relation. The equivalence classes for pc are called the
path components of x.
(b) Describe the set xpc for the following spaces:
i.x=S3
ii.x is the topologist's sine curve
iii. x=Z[0,1)
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