I would change XIX to 14th and change "in the end" to "at the end".
I'd like to ask for some help with my paper. I will post some of its fragments with some more comments. I'm not native speaker, but I'd like my text to be natural and I'd like it to be easy to read.
I'm not sure about:Homology theory is one of the main tools used in algebraic topology. We
can find its origins in the end of XIX century when people started their
research in order to understand manifolds and, more generally, simplicial
complexes. Later, by creating singular homology theory, the concepts have
been generalized to all topological spaces. Singular homology theory is well
understood for tame spaces (e.g. simplicial complexes, manifolds) and some
steps to grasp its behaviour for more complicated spaces has been taken by
a new emerging field which we call "wild algebraic topology" (it is focused
on spaces like: Hawaiian Earring [2] [3] [4], Harmonic Archipelago [6] [5],
Sierpinski-gasket [1], etc.)
1. "We can find its origins..."
2. "Later, by creating..." - maybe it should be "with creating" i don't know
3. I'm not sure about the tenses...
I would change XIX to 14th and change "in the end" to "at the end".
Thanks :D Is it really all? (19th, by the way ;))
All right, then. So I will post another paragraph. And I kindly ask for help with correcting all mistakes
The previously mentioned Hawaiian Earring whose first singular homol-
ogy group has been algebraically described in [4] is a sequence of circles
tangent to each other in one point with diameters converging to zero.
The reason behind why fairly simple space has a very complicated homology group
is that chains in singular homology consist of finite number of simplices, and
we clearly see that Hawaiian Earring has infinite number of "cycles".
difference between Hawaiian Earring and the join of countably many circles
is that it admits many singular simplices that wind around infinitely many
circles.
Hmm... Now i think that it should be "... a sequence of circles TANGENTIAL to each other AT one point". What do you think? Anybody?