#### HallsofIvy

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You are again referring to a "dense set" as if "dense" were a property of a single set. That's not true. Set A is "dense"I refer to the following defintions from the previously referenced Borowski and Borwein (1991 p 591) for "topology"

1. Point set topology: the branch of mathematics that is concerned with the generalization of the concepts of limits, continuity, etc to sets other than the real or complex numbers.

2. Algebraic topology: a branch of geometry describing the properties of a figure that are unaffected by continuous distortion such as stretching or knotting.

3. A family of subsets of a given set that constitute a topological space. The discrete topology consists of the entire power set, while the indiscrete topology contains only the empty set and the entire space.

I was only arguing that if T is the point set of a surface (ie the surface of a torus), T is dense in some other set. That is not to say that a topological space must contain a dense set.

**in set B**if and only B is equal to the closure of A.

I don't see WHY you arguing "that if T is the point set of a surface (ie the surface of a torus), T is dense in some other set." It's clearly not true. The surface of a torus, or any manifold with surface, is a closed set. B= closure of A cannot be true in that case unless B= A itself.

What were you referring morphism to?Because of the reference to "all subsets of A", it is clear that "all sets contain the empty set" should have been "all sets have the empty set as a subset". However, since it is the case that the poster's statement "If A is a dense set, then all subsets of A are dense" isfalsethe whole question becomes moot.