topological properties I have some questions about the initial and final topology, let $(f_i: X \rightarrow X_i)_{i \in I}$ be a source and let $(g_i: X_i \rightarrow X)_{i \in I}$ be a sink 
1) If $I$ is countable, and if all $X_i$ are first countable respectively second countable, does that imply that the initial topology on X (with source $f_i$ (see above)) is first countable respectively second countable?
2) If all $X_i$ is separable respectively compact, locally compact, connected or path connected and we have the final topology on $X$ (see $g_i$ in introduction), can anyone give me an example that $X$ is not separable respectively compact, locally compact, connected or path connected?
3) If $I$ is countable, and let $(f_i: X \to X_i)_{i\in I}$ a source with the initial topology on $X$. If all $X_i$ are (pseudo)metrizable does that imply $X$ is also (pseudo)metrizable? I think not but I can't find a counterexample.
 A: Proofs for 1) seem to be straightforward. Have you tried to prove it?
For 2) note that the topology on topological sum is final wrto inclusions of summands. This gives counterexaple to separability, compactness, connectedness and path connectedness. Do you assume being Hausdorff for locally compactness? If so then since there are non-Hausdorff quotients of locally compact Hausdorff spaces and quotient is another example of final topology, this gives a counterexample easily. If you don't assume that locally compact means Hausdorff that if I'm not wrong $\mathbb{R} / \mathbb{Q}$ in sense of quotient just contracting $\mathbb{Q}$ to a point is not locally compact in any sense (since any nbhd of $\mathbb{Q}$ contains infinite discrete closed subset).
For preserving of (pseudo)metrizability for initial topology: Any initial topology can be obtained as composition of following constructions: Taking products (in our example taking countable products), taking subspaces and taking initial topology generated by one quotient map. The first two cases preserves (pseudo)metrizability the third preserves pseudometrizability but not metrizability (e.g. quotient of two-point indiscrete space). So metrizability is not preserved but pseudometrizability is.
