TITLE: Why "singular" in "singular homology/cohomology"?
QUESTION [6 upvotes]: As the title suggests, I'm curious to know whether there is any reason why the word "singular" appears in "singular homology/cohomology".

REPLY [5 votes]: Because to obtain our chain complex for a space $X$ we look at arbitrary maps $\Delta^n \to X$ which are not necessarily nice, i.e. singular. In contrast we look at nice embeddings when dealing with simplicial homology.