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example of a connected space that is not
path-connected∗
yark†
2013-03-21 14:21:36
This standard example shows that a connected topological space need not
be path-connected (the converse is true, however).
Consider the topological spaces
X2
= {(0, y) | y ∈ [−1, 1]}
= (x, sin x1 ) | x > 0
X
= X1 ∪ X2
X1
with the topology induced from R2 .
X2 is often called the “topologist’s sine curve”, and X is its closure.
X is not path-connected. Indeed, assume to the contrary that there exists
a path γ : [0, 1] → X with γ(0) = ( π1 , 0) and γ(1) = (0, 0). Let
c = inf {t ∈ [0, 1] | γ(t) ∈ X1 } .
Then γ([0, c]) contains at most one point of X1 , while γ([0, c]) contains all of
X1 . So γ([0, c]) is not closed, and therefore not compact. But γ is continuous
and [0, c] is compact, so γ([0, c]) must be compact (as a continuous image of a
compact set is compact), which is a contradiction.
But X is connected. Since both “parts” of the topologist’s sine curve are
themselves connected, neither can be partitioned into two open sets. And any
open set which contains points of the line segment X1 must contain points of
X2 . So X is not the disjoint union of two nonempty open sets, and is therefore
connected.
∗ hExampleOfAConnectedSpaceThatIsNotPathconnectedi
created: h2013-03-21i by: hyarki
version: h33087i Privacy setting: h1i hExamplei h54D05i
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You can reuse this document or portions thereof only if you do so under terms that are
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