I find subsets, as presented in "foundations of analysis" type books, confusing. They start off with a definition of subset as "if x in A then x in B."

Later the discussion moves to subsets of a Euclidean space and open subsets. Is a surface S in R3 a subset of R3? What is the neighborhood of a subset of S? What is the interior of a subset of S? If the neighborhood is defined in R3, there is no such thing as an open surface as a subset in Euclidean space.

What implications does this have for a mapping from R3 to a subset of R3 if the subset is a surface?

In researching the question, I came up with the following:

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Atopological spaceis a setXtogether with a collectionTof subsets ofXsatisfying the following axioms:

The set

- The empty set and
Xare inT.- The union of any collection of sets in
Tis also inT.- The intersection of any pair of sets in
Tis also inT.Tis atopologyonX. The sets inTare theopen sets, and their complements inXare theclosed sets. The elements ofXare calledpoints.

A function between topological spaces is said to be

- More generally, the Euclidean spaces
Rnare topological spaces, and the open sets are generated by open balls.- Any metric space turns into a topological space if one defines the open sets to be generated by the set of all open balls.
continuousif the inverse image of every open set is open.

http://www.wordiq.com/definition/Topological_space

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which is a marvel of clarity. The only question remaining is, can you talk about continuity for a function which maps a "3-dimensional" subset of R3, say a solid sphere, into a line or surface in R3?