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A Connected Set is Pathwise Connected in a Metric Space

Last updated Nov 1, 2022

- Suppose not.

x, y

not Path-Connected. Create Sets of all points pathwise connected to

x

and those pathwise connected to

y

. This is an Open since each point has an Open Ball around it that is contained in the set. This creates a disconnection of the set.

Connected

Connected

Definition 1 Suppose $X$ is a . We say $X$ is if the only pair of s $U, V$ satisfying $U,...

11/7/2022
Metric Space

Metric Space

Definition A is a $M$ equipped with a $d: M \times M \to \mathbb{R}_{\geq0}$ . Remarks A is naturally endowed...

11/7/2022
Metric Topology

Metric Topology

Definition Let $(M, d)$ be a . The on $M$ is the topology generated by the basi $$\mathcal{B} = \{B_{\epsilon}(x) \subset...

11/7/2022