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FWIW, the one-point compactification of the discrete natural numbers can be described as follows: The underlying set is the natural numbers along with one extr
by kevinventullo 5y ago
FWIW, the one-point compactification of the discrete natural numbers can be described as follows:
The underlying set is the natural numbers along with one extra point which we’ll call infinity.
The open subsets are:
- Any subset which does not contain infinity,
- Any subset containing infinity whose complement is finite.
So, in order for an endomap to be continuous, it either has to
- Send infinity to infinity, in which case the pre-image of any natural must have finite cardinality (though the pre-image of infinity could be infinite).
- Send infinity to some natural number n, in which case all but finitely many naturals should map to n.
- Twisol 5y agoOh cool, thanks! > - Any subset which does not contain infinity We get those from the original (discrete) topology, I assume? > - Any subset containing infinity whose complement is finite. Ohh, is this construction related to the cofinite topology, then? I always wondered what that was useful for. (I never quite made it all the way through Munkres.) Also, I start to see why Wikipedia characterizes this as `{0} u {1/n | n in N}`. Any ball containing 0 will miss only finitely many points. > So, in order for an endomap to be continuous, [...] It took me a bit to work through this from first principles, but I see! Thanks for lighting the path.
- kevinventullo 5y ago> We get those from the original (discrete) topology, I assume? Yep! > ... is this construction related to the cofinite topology, then? I’m not sure, it’s sort of only cofinite “at infinity”? One place you see the cofinite topology appear is in algebraic geometry. The Zariski topology of a (connected) one-dimensional algebraic variety is the same as the cofinite topology. Glad I could help!