Post #1441656
2026-04-18 14:15 UTC
Suppose you "color in" some hypercubes in ℝⁿ, and look at a vertex. If the vertex's neighborhood is not all colored in or all not-colored-in, then there is some boundary between the colored and not-colored hypercubes. An interesting topological question is: is this boundary, in the neighborhood of the vertex, homeomorphic to an open ball in ℝⁿ⁻¹?
Replies (1)
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@jcreed@mastodon.social 2026-04-18 14:15
The thing I'm actually curious about is: Is this topological condition equivalent to to asserting that, for each dimension d, occupancy/colored-in-ness either always increases or decreses monotonically? If not, is there some other purely combinatorial condition that captures it?