For simplicial homology we supposed that our space had already been expressed as a simplicial complex, i.e., decomposed into a union of simplices. Some spaces cannot be expressed in such a way, and even those that can, can usually be expressed as a simplicial complex in many different ways. Choosing one way can obscure some details of the space. For these reasons “singular” homology was developed, which gets around this problem by, in a very loose sense, considering all possible simplicial decompositions.
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