example of a connected space that is not path-connected
is often called the “topologist’s sine curve”, and is its closure.
is not path-connected. Indeed, assume to the contrary that there exists a path (http://planetmath.org/PathConnected) with and . Let
Then contains at most one point of , while contains all of . So is not closed, and therefore not compact. But is continuous and is compact, so must be compact (as a continuous image of a compact set is compact), which is a contradiction.
But 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 must contain points of . So is not the disjoint union of two nonempty open sets, and is therefore connected.
|Title||example of a connected space that is not path-connected|
|Date of creation||2013-03-22 12:46:33|
|Last modified on||2013-03-22 12:46:33|
|Last modified by||yark (2760)|
|Defines||topologist’s sine curve|