(a) Let $\mathbf{x}$ and $\mathbf{y}$ be two points in the product. Let $t\in[0,1]$ and $\mathbf{z}(t)$ be a point such that $z_i(t)=f_i(t)$ where $f_i:[0,1]\rightarrow X_i$ is a path connecting $x_i$ with $y_i$ in $X_i$ . Since any open subbasis set $U$ in the product is an inverse projection $\pi_i^{-1}(U_i)$ of a proper open set $U_i\subset X_i$ for some $i$ , the preimage $\mathbf{z}^{-1}(U)$ is just $f_i^{-1}(U_i)$ which is open in $[0,1]$ . (The proof here repeats partially the proof of Theorem 19.6, which can be used here instead.)(b) No, the topologist’s sine curve is an example.(c) Yes, the composition of two continuous functions is a continuous function.(d) Take any two points. If they are in the same set in the collection, there is a path between them. If they are not in the same set in the collection then there is a path connecting the first point to a common point of all sets in the collection and another one connecting the common point to the second point, the joint path is still continuous and is a path connecting the point. I believe a more general statement is also true: if a path connected subspace has a common point with any set in a collection of path connected subspaces, then the union of the set with the union of the sets in the collection is path connected (the proof must be similar — the only difference should be that the constructed path may consist of three parts).