## Level Sets of Functions from Binary Trees

A quick lemma I found, while reading THE MORSE LANDSCAPE OF A RIEMANNIAN DISK by S. FRANKEL & M. KATZ. *Inst. Fourier, Grenoble 43, 2 (1993), 503-507*.

Let denote a binary tree of height . Recall, the binary tree of height zero is a point, the root, and that we construct by taking a point as its root and linking this point to the roots of two copies of .

If is a continuous function, then has a level set with points.

We proceed by induction. The cases and are easy, since we only require a single point in our level set. Consider and . By continuity of , and the closedness of , we can pick representatives such that and . As we are in a binary tree, we have a unique shortest path joining and . If this path passes through the root of then it must avoid a copy of sitting within . Otherwise, it avoids a copy of sitting within . Now consider the restriction of to the complement of this path. Since this induces a continuous function on the avoided binary tree, there must be some with pre-images. But then, considering on the minimal path, we have and hence we obtain another pre-image. It follows we have distinct pre-images of , completing the induction.

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