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location Fiji
age 37
visits member for 2 years, 7 months
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I'm a lecturer of mathematics at the University of the South Pacific. My research interests are in algebraic topology and metric geometry.


15h
comment Why do repeated linear factors have to be dealt with in this way?
is it a philosophical question? like why is $2+3\ne 4$?
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comment Why do repeated linear factors have to be dealt with in this way?
you say yourself that you know it doesn't work, and then you ask why you can't do that? Well, you can't do it because it does not work.
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answered Vector space and form?
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comment concepts which is present in metric space but not in topological space
Yes @KevinCarlson , a typical element p in \Omega(S), for any set S, is a collection of finite subsets of $S$ which is down closed, i.e., if a set is there, then any subset of it is there too.
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comment concepts which is present in metric space but not in topological space
arxiv.org/abs/1311.4940 is the note.
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revised concepts which is present in metric space but not in topological space
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answered concepts which is present in metric space but not in topological space
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comment concepts which is present in metric space but not in topological space
and just to be a bit more precise, completeness is a uniform property, not a metric one. And even more precise is that completeness is a Cauchy property, not a uniform one.
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comment concepts which is present in metric space but not in topological space
@AJY how do you propose to define a Cauchy sequence in a topological space?
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revised Is Cantor's diagonal argument dependent on the base used?
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comment Is Cantor's diagonal argument dependent on the base used?
yes, but it is not meant to be used as an interesting/efficient/useful manner to produce reals. In fact, the proof exploits a certain typographical property of the reals (their representations as strings of digits) to deduce something about cardinalities. The construction of a real not on the list depends on so many arbitrary choices ...
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comment Why do we first introduce the open set definition for continuity instead of the neighborhood definition?
@johndoe I refer more to Flagg's continuity spaces, which are based on ideas of Kopperman.
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comment Why do we first introduce the open set definition for continuity instead of the neighborhood definition?
@Hurkyl and I must agree to some extent with you too :) The question "what is topology" is the issue here. There are different models to saying what a topology is (e.g., in terms of open sets, closed sets, closure operators (which all give rise to isomorphic categories) or in terms of Flagg's continuity spaces (which gives an equivalent category)). As long as the categories are equivalent, we are speaking of different models for the same thing. So, which are the 'best' models? That depends of course. I personally like Flagg's continuity spaces. It's simple and intuitive.
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comment Is Cantor's diagonal argument dependent on the base used?
@KyleStrand and as soon as you bring probability into the picture you must get quite technical, much more than what I presented. Moreover, there is no such thing as a uniform probability in the sense you described.
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awarded  Nice Answer
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comment Is Cantor's diagonal argument dependent on the base used?
The cardinality of a list is countable, denoted by $\omega $. The cardinality of the reals can be shown not just to be not countable, but to be equal to $2^\omega$ (you can use binary representations again, you have two choices for each digit, and there are $\omega $ many digits to choose (ignoring double representations, though strictly speaking you need to account for these). Now, it can also be shown that $2^\omega - \omega = 2^\omega$. So, any listing of real numbers contains only $\omega $ many numbers, and misses $2^\omega$, as many as the entirely of all the reals.
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revised Is Cantor's diagonal argument dependent on the base used?
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revised Is Cantor's diagonal argument dependent on the base used?
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answered Is Cantor's diagonal argument dependent on the base used?
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comment Why do we first introduce the open set definition for continuity instead of the neighborhood definition?
I must agree to some extent with @PedroTamaroff at least in the following way. A topology is a rather large thing, typically consisting of a rather large cardinality of subsets and it is quite hard (if at all possible) to describe these sets directly, let alone understand them. In the presence of a metric one can describe the induced topology quite straightforwardly, and in the presence of a basis one typically obtains a much more manageable way to describing the topology. Both of these methods help in understand what topology is (and in fact, all topologies are metrizable, in a suitable way).