Given a space , there are many ways to define a topology. More common methods are one of the following:
- explicitly list out all the open sets
- define the rules for a subset to be open
- define the basis/subbasis of the topology (Note: The basis here refers to topological basis, this is different than a vector space basis that we have learnt in linear algebra.)
More often than not, we will work with topological spaces that has infinitely many open sets. In this case it will be hard to explicitly list out all the open sets in the topological space. In this post, we will look at a topology call cocountable topology, and luckily, we can describe the open set of this topology, thus we do not need to describe a topological basis.
Given a ground set , the cocountable topology on is the empty set together with the collection of subsets such that is countable, namely the complement of is countable. For example, if , then under the cocountable topology, none of the subsets of is open.
Now let’s assume that is an uncountable set. By definition, . Then, let be a collection of open sets in for some index set . Thus is countable for all . Now consider
by De Morgan’s law, where is arbitrary. Since is countable for all $\lambda \in \Lambda$ and intersections will only result in smaller set, therefore must be countable. Thus we may conclude that .
Finally, let be a finite collection. Then by De Morgan’s law, we have . Notice that finite union of countable sets is still countable (in fact, countable union of countable sets is countable), thus is countable and .
Therefore is indeed a topology.
Installing NVIDIA driver in Arch Linux is pretty simple. However, to set up the configurations can be frustrating. If you have just fresh installed Arch Linux, great! Just follow the following steps to install and configure NVIDIA drivers. Personally, I did not have a fresh Arch Linux, and hence I need to delete most of the graphics drivers and install them again.
Installing NVIDIA Driver
Personally, I prefer using
NVIDIA Optimus in conjunction with
Bumblebee (Yes, they are Transformers names. There is also a package for hybrid graphics call
Prime, check it out). To install the drivers after a fresh Arch Linux intallation, do the followin steps in that order.
pacman -S nvidia nvidia-lts nvidia-settings lib32-nvidia-utils lib32-nvidia-libgl mesa xf86-video-intel lib32-virtualgl
gpasswd -a <user> bumblebee
sudo systemctl enable bumblebeed.service
However, most people do not a fresh install. So we need to remove all the graphic drivers first. To do this, run either
pacman -Rdd nvidia nvidia-lts nvidia-settings lib32-nvidia-utils lib32-nvidia-libgl mesa xf86-video-intel lib32-virtualgl
pacman -Rs nvidia nvidia-lts nvidia-settings lib32-nvidia-utils lib32-nvidia-libgl mesa xf86-video-intel lib32-virtualgl
Either one of the above should work. Warning: If all else fails, you can try
pacman -Rsc nvidia nvidia-lts nvidia-settings lib32-nvidia-utils lib32-nvidia-libgl mesa xf86-video-intel lib32-virtualgl, but I am strongly against this command unless you know what you are doing.
To test that the driver is running correctly, install
mesa-demos and use
glxgears to test if it works by running
optirun glxgears -info. If that fails, try running
optirun glxspheres64 for 64 bit system, or
optirun glxspheres32 for 32 bit system.
Running a Program with NVIDIA
optirun in front of the commands. For example, if we want to run
app.exe, simply type in
optirun wine app.exe in the terminal.