2
votes
Accepted
Definition clarification on topology regarding closed set
"Is (−∞,a)∪(b,∞) open just because it is union of open intervals?"
Yes. By definition, the topology $T$ on $\mathbb R$ is the set whose elements are the unions of open intervals.
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2
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Definition clarification on topology regarding closed set
It follows from the definition of topology and from the definition of the usual topology on $\mathbb{R}$.
The usual topology on $\mathbb{R}$ is generated by the basis of open intervals $\{]a, b[ \mid ...
- 572
2
votes
Definition of Riemann integral using $\displaystyle\lim_{N\to +\infty} \sum_{i=1}^N f(x_i)(x_{i+1} - x_i)$
Koren Parkhov's answer hit on one of your points, the case of "degenerate" rectangles. That is resolved by assuming that a partition takes the form
$$
x_0 = a < x_1 < x_2 < \cdots &...
- 47.4k
2
votes
Definition of Riemann integral using $\displaystyle\lim_{N\to +\infty} \sum_{i=1}^N f(x_i)(x_{i+1} - x_i)$
The classical definition assumes $x_0<x_1<...<x_n$ . indeed, if we allow the partition you suggested, there is a problem and the limit might not exist.
1
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Is the category of small $\mathcal{U}$-categories a category? (According to Kashiwara and Schapira's conventions)
Yes, I agree that if a category has a set of objects and you want a category of $\mathcal U$-small categories, you will need to insist that your small categories have a genuine $\mathcal U$-set of ...
- 53.1k
1
vote
Accepted
Definition of Riemann integral using $\displaystyle\lim_{N\to +\infty} \sum_{i=1}^N f(x_i)(x_{i+1} - x_i)$
There are a number of tiny inconsistencies:
first of all, when you define $N$ point, you only have $N-1$ intervals, and for $i=N$, $x_{i+1}$ is not defined: the sum should only be $\lim_{N\to \infty} ...
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