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Number of dense and open subsets of $\Bbb R$
Can someone tell me the following question:
Suppose $\Bbb R$ is endowed with the usual topology (i.e. the usual topology, just on $\Bbb R$, not on the set $\mathbb R \times \mathbb R$).
1) What is the number of dense subsets of $\Bbb R$?
2) What is the number of open subsets of $\Bbb R$?
I don’t have any general intuition about this. My friend thinks that there are $2^{\aleph_0}$ of each kind.
A:
The number of open dense subsets of $\Bbb R$ is $2^{2^{\aleph_0}}$, since every open dense subset of $\Bbb R$ is homeomorphic to $\Bbb R$ itself.
(As an aside, there are many more open discrete subsets of $\Bbb R$ than dense ones, since we may have $\sup\{x_\alpha\mid \alpha\sup\{x_\alpha\mid \alpha https://cryptobizworld.news/spherical-panorama-fisheye-polar-video-writer-crack-free-obtain-mac-win-latest-2022/
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I am finishing my first year as Director of the Institute of Education, University of St. Andrews, Scotland. It is a pleasure to be writing this report for the staff and students of the IEO. I am, of course, happy that we have ended the year on an upwards spiral of success, praise and congratulations. To write a summary report is a daunting task and here I have had to attempt to draw a balance. I have had to try to sum up our work so far in a way that makes it easy to identify our achievements and define our limitations.
I came to the IEO as Director in May 2009, from the Institute for the Study of European Economic History, where I had been Director for five years. In addition to my professional responsibility of “management”, I was also responsible for running a research group at that Institute. So my first year at the IEO has been just as busy, on a smaller scale.
In St Andrews, we have a small team of four full time Research Assistants (two in information communication, two in economics), all of whom have earned their place at the Institute through outstanding research, teaching and administrative excellence. To say we have high standards is an understatement!
We have a Research Committee of ten people – five academics and five industry professionals. The committee meets to review the research programmes and discuss practical issues for the development of those programmes. Recently, we have had the opportunity to review our research priorities and have defined the following lines of research in the first two years:
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