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Homogeneous spaces generalized – and an open problem
Homogeneous spaces are topological spaces that locally look everywhere the same. Put in mathematical terms, is homogeneous if for any there is an isomorphism such that . We can generalize this by not restricting ourselves to a pair of points … Continue reading
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Two notions of continuity in the space of natural sequences
In this presentation by Martín Escardó I came across a nice concept of continuity in the set of all functions . For and we say that if and only if for all we have . Then we define to be finitely … Continue reading
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A reformulation of Hadwiger’s conjecture
Let be a finite, simple, undirected graph. Hadwiger’s conjecture states that if then is a minor of . Let’s call that statement (Hadw). We show that it is equivalent to the following statement: (Hadw’): If is not a complete graph, … Continue reading
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Measuring subsets of N
Recently I tried to put a measure on the powerset of the natural numbers , but it was pointed out in a comment that the map I defined is not even finitely additive. Question: Is there a map with the … Continue reading
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Additive?
Recently I was trying to put a “meaningful” measure on the set of natural numbers. The first idea was to measure the density of a subset on natural intervals and take the limit of it, that is However, the limit … Continue reading
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Exceptions to the Friendship Paradox
The Friendship Paradox says that your friends are likely to have more friends on average than you do. I was wondering whether there are settings in which the statement of the Friendship Paradox does not hold. Below we think about … Continue reading
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Twodimensional convergence
Here is something that I was certain would have an easytoprove positive answer; but it turns out that there is an exceedingly simple counterexample. Let be the first infinite ordinal; its successor is .We consider a map , or put differently, a Matrix … Continue reading
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