The precise definition of (finite) linear graph?

In Munkres' book "Topology", (finite) linear graph is defined in P.308, Example 6, as the following picture shows.

http://i3.6.cn/cvbnm/0d/fb/12/f8c282...71c4c83130.jpg

The implication from compact to closed for edges demands that these edges be regarded as compact subspaces of , which, however, is not mentioned in the definition of (finite) linear graph in this paragraph. I searched the Internet but did not find its precise definition, so I hope I can find help here: what is the precise definition of (finite) linear graph? Is it indispensable that each edge, or arc, is a subspace of ?

Another question: if we union a number of spaces together, can the union be topologized such that each is a subspace of the union space ?

Thanks!