Strong Topology is the strongest topology? In his article Construction of universal bundles. II (1956), John Milnor defines the strong topology in a join of spaces, but his definition is

By a strong topology in $A_1\circ A_2\circ \dots \circ A_n$ we mean the strongest topology such that the coordinate functions
$t_i\colon A_1\circ A_2\circ \dots \circ A_n\longrightarrow [0,1] $  and $a_i\colon t_i^{-1}(0,1]\longrightarrow A_i$
are continuous.

I don't understand the definition of his strong topology. Could anyone explain me it (if possible with an example)? Another question too is: what does he mean with the "strongest topology"?
 A: What Milnor calls the strong topology on the join 
$$X= X_1 * \cdots * X_n  $$
is called the join topology in Topology and Groupoids (T&G) Section 5.7, and is the initial topology with respect to the functions he gives. More formally, this means that a function $f: Z \to X$ is continuous if and and only if its composites with all the "coordinate functions" $t_i,a_i$ are continuous. So this topology is well placed for deciding if functions to the join are continuous. This is discussed in Section 5.6 of T&G. 
A little care is needed in discussing initial topologies in this case since the functions $a_i$ are partial functions on the join $X$, that is they are not defined on all of $X$. 
More generally, if $X$ is a set and $f_i: X \to X_i$ is a family of partial functions to topological spaces $X_i$, then the initial topology on $X$ with respect to the $f_i$ is the smallest (= coarsest) topology such that all the $f_i$ are continuous. 
One of the advantages of this topology is that it is easy to prove associativity of the join, which may not be so if you use an identification (= final) topology, as is quite common. 
