I've just started reading about algebraic topology. I've read the definitions of a $\Delta$-complex structure.
I have a couple questions:
How would one construct one for $S^1$? I can see that if we take a 1-simplex and identify its end points, does that work? Also I'm not sure how this can be written explicitly.
Secondly, why are simplicial and $\Delta$ complex structures defined? Is one used in a particular situation where the other isn't?