continuous images of path connected spaces are path connected
Proposition.
The continuous image of a path connected space is path connected.
Proof.
Let X be a path connected space, and suppose f is a
continuous surjection whose domain is X. Let a and b
be points in the image of f. Each has at least one preimage
in
X, and by the path connectedness of X, there is a path in
X from a preimage of a to a preimage of b. Applying
f to this path yields a path in the image of f from a
to b.
∎
| Title | continuous images of path connected spaces are path connected |
|---|---|
| Canonical name | ContinuousImagesOfPathConnectedSpacesArePathConnected |
| Date of creation | 2013-03-22 15:52:38 |
| Last modified on | 2013-03-22 15:52:38 |
| Owner | mps (409) |
| Last modified by | mps (409) |
| Numerical id | 6 |
| Author | mps (409) |
| Entry type | Result |
| Classification | msc 54D05 |