Initial topology

In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set $X$, with respect to a family of functions on $X$, is the coarsest topology on ''X'' which makes those functions continuous function (topology). ==Definition== Given a set ''X'' and a family (mathematics) of topological spaces ''Y''_''i'' with functions :$f\_i:\; X\; \backslash to\; Y\_i$ the initial topology τ on $X$ is the coarsest topology such that each :$f\_i:\; (X,\backslash tau)\; \backslash to\; Y\_i$ is continuous function (topology). Explicitly, the initial topology may be described as the topology subbase sets of the form $f\_i^\{-1\}(U)$, where $U$ is an open set in $Y\_i$. ==Examples== Several topological constructions can be regarded as special cases of the initial topology. * The subspace topology is the initial topology on the subspace with respect to the inclusion map. * The product topology is the initial topology with respect to the family of projection maps. * The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms. * The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space. * Given a family (mathematics) of topologies {τ_''i''} on a fixed set ''X'' the initial topology on ''X'' with respect to the functions id_''X'' : ''X'' → (''X'', τ_''i'') is the supremum (or join) of the topologies {τ_''i''} in the lattice of topologies on ''X''. That is, the initial topology τ is the topology generated by the union (set theory) of the topologies {τ_''i''}. ==Properties== The initial topology on ''X'' can be characterized by the following universal property: a function $g$ from some space $Z$ to $X$ is continuous if and only if $f\_i\; \backslash circ\; g$ is continuous for each ''i'' ∈ ''I''. By the universal property of the product topology we know that any family of continuous maps ''f''_''i'' : ''X'' → ''Y''_''i'' determines a unique continuous map :$f\backslash colon\; X\; \backslash to\; \backslash prod\_i\; Y\_i\backslash ,$ If the family of maps {''f''_''i''} ''separates points'' in ''X'' (i.e. for all ''x'' ≠ ''y'' in ''X'' there exists some ''f''_''i'' such that ''f''_''i''(''x'') ≠ ''f''_''i''(''y'')) then the map ''f'' will be a topological embedding if and only if ''X'' has the initial topology determined by the maps ''f''_''i''. In the language of category theory, the initial topology construction can be described as follows. Let ''Y'' be a functor from a discrete category ''J'' to the category of topological spaces Top which selects the spaces ''Y''_''i'' for ''i'' in ''J''. Let Δ be the diagonal functor from Top to the functor category Top^''J'' (this functor sends each space ''X'' to the constant functor to ''X''). The comma category (Δ ↓ ''Y'') is then the category of all cones to ''Y'', i.e. objects in (Δ ↓ ''Y'') are pairs (''X'', ''f'') where ''f''_''i'' : ''X'' → ''Y''_''i'' is a family of continuous maps on ''X''. If ''U'' is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^''J'' then the comma category (Δ′ ↓ ''UY'') is the category of all cones to ''UY''. The initial topology construction can then be described as a functor from (Δ′ ↓ ''UY'') to (Δ ↓ ''Y''). This functor is adjoint functors to the corresponding forgetful functor. == See also == * Final topology Topology

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