example of a connected space that is not path-connected

This standard example shows that a connectedPlanetmathPlanetmath topological spaceMathworldPlanetmath need not be path-connected (the converseMathworldPlanetmath is true, however).

Consider the topological spaces

X1 ={(0,y)y[-1,1]}
X2 ={(x,sin1x)x>0}
X =X1X2

with the topology induced from 2.

X2 is often called the “topologist’s sine curve”, and X is its closureMathworldPlanetmathPlanetmath.

X is not path-connected. Indeed, assume to the contrary that there exists a path (http://planetmath.org/PathConnected) γ:[0,1]X with γ(0)=(1π,0) and γ(1)=(0,0). Let


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 compactPlanetmathPlanetmath. But γ is continuousPlanetmathPlanetmath 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 contradictionMathworldPlanetmathPlanetmath.

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 unionMathworldPlanetmath of two nonempty open sets, and is therefore connected.

Title example of a connected space that is not path-connected
Canonical name ExampleOfAConnectedSpaceThatIsNotPathconnected
Date of creation 2013-03-22 12:46:33
Last modified on 2013-03-22 12:46:33
Owner yark (2760)
Last modified by yark (2760)
Numerical id 16
Author yark (2760)
Entry type Example
Classification msc 54D05
Related topic ConnectedSpace
Related topic PathConnected
Defines topologist’s sine curve