Definition:Region/Metric Space

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Let $M = \left({A, d}\right)$ be a metric space.

A region of $M$ is a subset $U$ of $M$ such that $U$ is:

$(1): \quad$ non-empty
$(2): \quad$ path-connected.

Also defined as

Some sources insist that in order for a subset of a metric space to be a region it must also be open.