Definition:Michael's Product Topology
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Definition
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers.
Let $\struct {S, \sigma} := \struct {\R, \tau^*} \times \struct {\Bbb I, \tau'}$, where:
- $\tau^*$ is the discrete irrational extension of $\tau_d$ by $\Bbb I$
- $\tau'$ is the subspace topology on $\Bbb I$ induced by $\tau_d$.
$\struct {S, \sigma}$ is referred to as Michael's product topology.
Also see
- Results about Michael's product topology can be found here.
Source of Name
This entry was named for Ernest Arthur Michael.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $85$. Michael's Product Topology