2. the sum of zero terms (the empty sum) is 0; the product of zero terms (the empty product) is 1
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3. the degree of the zero polynomial is -∞
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4. the maximum of zero numbers is -∞; the minimum of zero numbers is +∞
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5. the determinant of the 0x0 matrix (the empty matrix, with no entries) is 1
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6. the empty graph is disconnected, and so is the empty topological space, for the same reason that 1 isn't a prime number
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7. there's a different empty manifold in each dimension: a zero-dimensional empty manifold, a 1-dimensional empty manifold, etc. (and of course none of them are connected or path-connected)
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That's an unreasonable convention. The dimension of a manifold is a locally constant function on the manifold.
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you need a different empty manifold in each dimension for it to always be true that an n-manifold with boundary has a boundary which is an (n-1)-manifold, and in turn you need this to always be true to make cobordism categories work
ncatlab.org/nlab/show/cobo
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one way to say this is that "n-manifold" fundamentally includes n as part of the definition; the fact that invariance of domain lets you recover n from the underlying topological space for non-empty manifolds is a convenience but it obscures what's going on
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Is the disjoint union of two differently-dimensioned manifolds a manifold
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i actually know of no applications of allowing this. everywhere i’ve seen manifolds show up in other branches of math (most prominently in the neighborhood of the cobordism hypothesis, quantum topology, etc) we only ever consider n-manifolds for some n
here’s one simple observation that’s related: if you allow disconnected manifolds with components of different dimensions then you can’t consider the tangent bundle of a smooth manifold to be a principal G-bundle for any fixed G
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*classified by a principal G-bundle
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In algebraic geometry, it's just a necessity. And it's a good habit to allow for analytification functors to exist in the greatest possible generality, I believe.
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