Definition:Compatible Quasiuniformities
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Definition
Let $\UU_1$ and $\UU_2$ be quasiuniformities on a set $S$.
Let $\struct {\struct {S, \UU_1}, \tau_1}$ and $\struct {\struct {S, \UU_2}, \tau_2}$ be the quasiuniform spaces generated by $\UU_1$ and $\UU_2$.
Then $\UU_1$ and $\UU_2$ are compatible (with each other) if and only if their topologies are equal.
That is, if and only if $\tau_1 = \tau_2$.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Uniformities