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... Proof. Suppose X is first countable, and A ⊆ X has the property that, if C is any compact set in X, the set A ∩ C is closed in C. We want to show tht A is closed in X. Since X is first countable, this is equivalent to showing that any sequence (xi ) in A converging to x implies that x ∈ A. Let C = { ...
... Proof. Suppose X is first countable, and A ⊆ X has the property that, if C is any compact set in X, the set A ∩ C is closed in C. We want to show tht A is closed in X. Since X is first countable, this is equivalent to showing that any sequence (xi ) in A converging to x implies that x ∈ A. Let C = { ...
Topology - Homework Sets 8 and 9
... number of polynomials p1 , . . . , pK . Note that (π, 0) is not in Q × {0}. However, there is a sequence ( xn )n of rational numbers with the property that xn converges to π in the standard topology (e.g., take xn to be the n place decimal expansion of π). Then ( xn , 0) ∈ Q × {0}, so f k ( xn , 0) ...
... number of polynomials p1 , . . . , pK . Note that (π, 0) is not in Q × {0}. However, there is a sequence ( xn )n of rational numbers with the property that xn converges to π in the standard topology (e.g., take xn to be the n place decimal expansion of π). Then ( xn , 0) ∈ Q × {0}, so f k ( xn , 0) ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.