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<li><math>F</math> is a vector subspace of <math>Y^T = \prod_{t \in T} Y,</math><ref group=note>Because <math>T</math> is just a set that is not yet assumed to be endowed with any vector space structure, <math>F \subseteq Y^T</math> should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined.</ref> which denotes the set of all <math>Y</math>-valued functions <math>f : T \to Y</math> with ___domain <math>T.</math></li>
</ol>
===𝒢-topology===▼
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
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<math display="block">\mathcal{U}(G, N) := \{f \in F : f(G) \subseteq N\}.</math>
▲===𝒢-topology===
▲Then the family
<math display="block">\{ \mathcal{U}(G, N) : G \in \mathcal{G}, N \in \mathcal{N} \}</math>
forms a [[Neighbourhood system|neighborhood basis]]<ref>Note that each set <math>\mathcal{U}(G, N)</math> is a neighborhood of the origin for this topology, but it is not necessarily an ''open'' neighborhood of the origin.</ref>
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