Initial Topology - Examples

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 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 of the topologies {τ_i}.
• A topological space is completely regular if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions.
• Every topological space X has the initial topology with respect to the family of continuous functions from X to the Sierpiński space.