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.

$\blacksquare$


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