how do you say this theorem? My attempt:
Let S be a Topological space, and let T-naught and the sets indexed by w be connected subsets of S. Assume T-naught and the sets indexed by w are not disjoint for each w in W. Then the union of T-naught and the union of elements of T sub w is connected
For $T_0$ I might say "Tee sub zero" or "Tee naught" depending on the phase of the moon. Going with "Tee naught" today, here might be a way I'd say the theorem (with adjustments to be understandable without the aid of notation)
"Let Ess be a topological space, and let Tee-naught and a collection of Tee-sub-double-u (indexed by double-u) be connected subsets of Ess. Assume Tee-naught and Tee-sub-double-u intersect nontrivially for each index double-u. Then the union of Tee-naught and all the Tee-sub-double-u is connected."
In yours, you say "the union of elements of T sub w", which is not quite right: you want to union the $T_w$'s themselves, not the elements of a particular $T_w$.