Fundamentalgruppen af plane mængder - Københavns Universitet

2 Resume
This paper is a Master Thesis written at the University of Copenhagen. The
subject is the fundamental group of planar subsets. The main result is that the
fundamental group of a planar subset injects into the first čhech group.
The first section is an introduction to the inverse systems over an abitrary category.
The second section deals with the important notion of a peano space. Here we
prove that a peano space is locally and globally path connected, and that the
image of the unit interval into a Hausdorff space is a peano space.
The third section is about planar sets especially planar peano space. When dealing
with planar peano spaces the Sierpinski carpet turns out to be very important.
We therefore finish this section with some results concerning the Sierpinski carpet
and partially filled sierpinski carpets.
In the fourth section we introduce the notion of the nerve of an open cover of a
topological space, and we use this to define the čhech group. We then prove the
main theorem of the thesis.
The last section deals with a consequence of our main theorem. This involves
a little bit of group theory. There is also an example that shows that the main
theorem doesn’t hold in R 3 .
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