Hatcher's Algebraic Topology - CW complexes Why isnt the topology of the CW complex a metric topology when there is infinitely many edges from a vertex, but it is when there are finitely many edges from a vertex? - page 83
 A: 
1- if the sequences of points inside of the edges do not converge to
  the vertex in the first place then why does it matter if there are
  infinitely or finitely many edges from it.

Hatcher says that "a sequence of points in the interiors of distinct edges ... never converges". If there are only finitely many edges, you can't get an infinite sequence of points in distinct edges. You could have an infinite sequence of points on a single edge converging to some point on that edge.

2- a metric topology should have a neighborhood around a vertex with
  infinitely or finitely many edges.

I'm not sure what the question is.
Imagine that you have an infinite set of edges $I_n$, $n \geq 1$, all sharing a vertex $v$. Give each edge the usual metric topology, and I'll write $d$ for the metric on each edge. With the weak topology on the union, the following is an open neighborhood of $v$: take the union over $n$ of the points $x$ on $I_n$ which satisfy $d(x,v) < 1/n$. This will not be open in the metric topology on the union, since it does not contain any $\epsilon$-ball around $v$.
