Definition of a (topologically) continuous functor In Atiyah's K-theory page 6 (http://www.cimat.mx/~luis/seminarios/Teoria-K/Atiyah_K_theory_Advanced.pdf), he defines a covariant functor of one variable $T$ from finite dimensional vector spaces to finite dimensional vector spaces to be continuous if the map $T:\hom(V,W)\to \hom(T(V),T(W))$ is continuous (with the topology of hom being that of $\mathbb R^n$). Then he develops some theory and claims this works too for functors of more than one variable, although he doesn't give an explicit definition of continuity in this case. What I think a possible definition could be is (eg, 2 variables):
$T$ is continuous if the map $T:\hom(V_1,W_1)\times\hom(V_2,W_2)\to\hom(T(V_1,W_1),T(V_2,W_2))$ is continuous, or if $T:\hom(V_1,W_1)\times\hom(V_2,W_2)\to\hom(T(V_1,V_2),T(W_1,W_2)) $. 
Alternatively, $T$ is continuous if $T:\hom(V_1,W_1)\times\hom(V_2,W_2)\to\hom(T(V_1),T(W_1))\times \hom(T(V_2),T(W_2))$ continuous. 
Lastly, it could also be continuity of $T:\hom(V_1,W_1)\times\hom(V_2,W_2)\to\hom(T(V_1),T(W_1))\otimes \hom(T(V_2),T(W_2))$.
I think the latter is the actual proper one.
 A: The definition mirrors that which would become enriched category theory https://en.wikipedia.org/wiki/Enriched_category (if I have my chronology correct), the base for enrichment being the Cartesian closed symmetric monoidal category $Top$, of say compactly generated Hausdorff spaces. In this context a continuous functor is simply an enriched functor.
The category with which we are working is $_\mathbb{K}Vect$ of finite dimensional $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$ vector spaces. The hom-spaces $hom(V,W)$ are topologised as Stiefel manifolds, and this is defined by choosing bases for $V$, $W$ to give isomorphisms $V\cong\mathbb{R}^n$, $W\cong\mathbb{R}^m$ and using these to pull back the Euclidean topology from $hom(\mathbb{R}^n,\mathbb{R}^m)\cong\mathbb{R}^{mn}$. The resulting topology should be independent of the chosen bases.
Now the product $\mathcal{C}\times\mathcal{D}$ of two categories $\mathcal{C}$, $\mathcal{D}$ enriched over the same symmetric monoidal category $(\mathcal{V},\otimes)$ exists. It has objects the pairs $(c,d)$ with $c\in ob(\mathcal{C})$,$d\in ob(\mathcal{D})$, and hom-objects $(\mathcal{C}\times\mathcal{D})((c,d),(c',d'))=\mathcal{C}(c,c')\otimes\mathcal{D}(d,d')$ (these are objects in $\mathcal{V}$). The composition and identity morphisms are not difficult to write down.
Taking $\mathcal{C}=\mathcal{D}={_\mathbb{K}Vect}$ and $(\mathcal{V},\otimes)=(Top,\times)$, where $\times$ is the categorical product, we get the $n$-fold product with objects $n$-tuples of vector spaces $(V_1,\dots,V_n)$, $(W_1,\dots,W_n)$, and hom spaces
$hom(V_1,W_1)\times hom(V_2,W_2)\times\dots\times hom(V_n,W_n)$.
(here $hom$ is the standard Stiefel manifold as previously discussed). Now continuity (enrichment) of a functor $\prod^n{_\mathbb{K}Vect}\xrightarrow{T}{_\mathbb{K}Vect}$ from the $n$-fold product, is the requirement that the induced map
$hom(V_1,W_1)\times\dots\times hom(V_n,W_n)\xrightarrow{T} hom(T(V_1,\dots,V_n),T(W_1,\dots,W_n))$
be continuous for all $V_1,\dots V_n,W_1,\dots W_n$.
Since we are working over $Top$, the product to consider is the categorical product, with which the monoidal structure on $Top$ is defined, and not the tensor product of vector spaces. Observe that we have chosen $Top$ to be a 'convenient category of topological spaces' to ensure that the adjunction maps $Map(X\times Y,Z)\cong Map(X,Map(Y,Z))$ are continuous. This is one important feature that is our motivation for restricting from the category of all topological spacs.
