On the Scott topology on the set \(C(Y,Z)\) of continuous maps. (English) Zbl 0789.54022
Summary: It is shown, that if a topology \(t\) contains the topology of pointwise convergence and it is splitting on the set \(C(Y,Z)\) of continuous maps, then the specialization order of \(t\) coincides with the pointwise order induced on \(C(Y,Z)\) by the specialization order of \(Z\).
This result is used to prove that when there exists the coarsest jointly continuous topology on \(C(Y,Z)\), where \(Z\) is an injective \(T_ 0\) topological space, then this topology is the Scott topology \(\sigma(C(Y,Z))\), which is determined by the pointwise order induced on \(C(Y,Z)\) by the specialization order of \(Z\).
This result is used to prove that when there exists the coarsest jointly continuous topology on \(C(Y,Z)\), where \(Z\) is an injective \(T_ 0\) topological space, then this topology is the Scott topology \(\sigma(C(Y,Z))\), which is determined by the pointwise order induced on \(C(Y,Z)\) by the specialization order of \(Z\).
MSC:
| 54C35 | Function spaces in general topology |
Keywords:
topology of pointwise convergence; specialization order; pointwise order; jointly continuous topology; Scott topologyReferences:
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