# kification　`prop.cg-coreflective`
English entry · root `../`
[in graph](../#n=prop.cg-coreflective)　·　[Chinese page (authoritative)](./prop.cg-coreflective.md)

> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 18 · proposition · cluster 紧生成空间与弱 Hausdorff (`topo.cg`) · Topology

## Abstract
The category $k\mathbf{Top}$ of compactly generated spaces is a coreflective subcategory of $\mathbf{Top}$: the inclusion $i$ has a right adjoint $k$ that keeps the underlying set and replaces the topology by the $k$-topology, $i \dashv k$, i.e. $\operatorname{Hom}_{\mathbf{Top}}(X, Y) \cong \operatorname{Hom}_{k\mathbf{Top}}(X, kY)$ for compactly generated $X$. The counit $kY \to Y$ is the identity map, continuous because the $k$-topology is the finer one.

> 续见 nodes/prop.cg-coreflective.en.2.md