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... on X is initial with respect to the family of mappings fi : X → Xi , i ∈ I, if T is the coarsest topology on X which makes all fi ’s continuous. The initial topology is characterized by the condition that a map g : Y → X is continuous if and only if every fi ◦ g : Y → Xi is continuous. Sets S = {fi− ...
... on X is initial with respect to the family of mappings fi : X → Xi , i ∈ I, if T is the coarsest topology on X which makes all fi ’s continuous. The initial topology is characterized by the condition that a map g : Y → X is continuous if and only if every fi ◦ g : Y → Xi is continuous. Sets S = {fi− ...
A NEW TOPOLOGY FROM AN OLD ONE Halgwrd Mohammed
... space X are denoted by ωδ O(X) (resp. ωδ C(X)). It is easy to see that each clopen, θ-open, ωθ -open, regular open and δ-open sets are ωδ -open and each ωδ -open set is ω o -open and ω-open, but not conversely. The following examples support our claim. Example 3.2. Let X={a, b} and τ ={φ,{a},X}. The ...
... space X are denoted by ωδ O(X) (resp. ωδ C(X)). It is easy to see that each clopen, θ-open, ωθ -open, regular open and δ-open sets are ωδ -open and each ωδ -open set is ω o -open and ω-open, but not conversely. The following examples support our claim. Example 3.2. Let X={a, b} and τ ={φ,{a},X}. The ...
Spaces of functions
... Boundedness The case when X is a compact metric spaces The Arzela-Ascoli theorem The compact-open topology ...
... Boundedness The case when X is a compact metric spaces The Arzela-Ascoli theorem The compact-open topology ...