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It is well known that for many semilinear parabolic equations there is a global attractor which has a cell complex structure with finite dimensional cells. Additionally, many semilinear parabolic equations have equilibria with finite dimensional unstable manifolds. In this article, these results are unified to show that for a specific parabolic equation on an unbounded domain, the space of heteroclinic orbits has a cell complex structure with finite dimensional cells. The result depends crucially on the choice of spatial dimension and the degree of the nonlinearity in the parabolic equation, and thereby requires some delicate treatment.