# Hall's Theorem for infinite graphs (Compactness theorem)

To Proof:

Let $$G = (V,E)$$ be an infinite bipartite graph with $$V = S \overset{.}{\cup}T$$ and finite node degree for each node.

G has a matching, that covers a set S iff for all subset $$H \subseteq S: |N(H)| \geq |H|$$ where $$N(H)= \{x \in V \mid \exists y \in H \text{ that } \{x,y\} \in E\}$$

This is hall's theorem with only difference of infiniteness of the bipartite graph.

How can I proof this by using Compactness theorem and Hall's theorem?

• What have you tried? Do you know how topology comes to the picture? And which topology to use? – Blackbird Apr 22 at 12:45