Having shown and known that:
And having verified that the tower that results from our work on problem 5 has the surjective splitting property.
And Upon agreeing on that:
We want to verify that the coordinate functions determine the topology on $\mathcal{L};$ that is a function $g : Z \rightarrow \mathcal{L}$ is continuous iff $p_{n} \circ g$ is continuous for each n.
Could anyone help me in proving so, please?
This holds for all inverse limits, by the transitive law of initial topologies that I showed here: the product has the initial topology wrt the projections and the inverse limit has the initial topology wrt the canoncial injection (subspace topology) into that product, so the inverse limit has the initial topology wrt the restriction of the projections ($p_n$, here) and then the universal mapping theorem for initial topologies finishes the proof.
Just abstract theory will already give you this.
