Equivalence of Definitions of Connected Manifold
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Theorem
Let $M$ be a topological manifold.
The following definitions of the concept of Connected Manifold are equivalent:
Definition 1
$M$ is called a connected manifold if and only if $M$ is a connected topological space.
Definition 2
$M$ is called a (path-) connected manifold if and only if $M$ is a path-connected topological space.
Proof
Definition 1 implies Definition 2
Let $M$ be connected.
From Topological Manifold is Locally Path-Connected:
- $M$ is locally path-connected.
From Connected and Locally Path-Connected Implies Path Connected:
- $M$ is path-connected.
$\Box$
Definition 2 implies Definition 1
Follows immediately from Path-Connected Space is Connected.
$\blacksquare$
Sources
- 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.): Chapter $4.$ Connectedness and Compactness