Path-connected implies connected
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., path-connected space) must also satisfy the second topological space property (i.e., connected space)
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Further information: connected space
A topological space is termed connected if it cannot be expressed as a disjoint union of two nonempty open subsets.
Further information: path-connected space
The converse is not true, i.e., connected not implies path-connected.
However, it is true that connected and locally path-connected implies path-connected.
The key fact used in the proof is the fact that the interval is connected. The proof combines this with the idea of pulling back the partition from the given topological space to .
Given: A path-connected topological space .
To prove: is connected.
Proof: We do this proof by contradiction. Suppose is not connected. Then, there exist nonempty disjoint open subsets such that . Pick a point and a point .
By assumption, there exists a continuous function such that and . Consider the subsets and . These are disjoint in and their union is . By the continuity of , they are both open in . Finally, since and , they are both nonempty. We have thus expressed as a union of two disjoint nonempty open subsets, a contradiction to the fact that is connected. This completes the proof.