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The path spaces of a directed graph play an important role in the study ofgraph $\css$. These are topological spaces that were originally constructedusing groupoid and inverse semigroup techniques. In this paper, we develop asimple, purely topological, approach to this construction, based on Tychonoff'stheorem. In fact, the approach is shown to work even for higher dimensionalgraphs satisfying the finitely aligned condition, and we construct the groupoidof the graph. Motivated by these path space results, we prove a Tychonofftheorem for an infinite, countable product of locally compact spaces. The mainidea is to include certain finite products of the spaces along with theinfinite product. We show that the topology is, in a reasonable sense, apointwise topology.

Tychonoff's theorem for locally compact space and an elementary approach to the topology of path spaces