Lecture 5 and 6
... A subnet of a net (xi : i ∈ I) is a net (yj : j ∈ J), together with a map j → # ij from J to I, so that xij = yj for all j ∈ J for all i0 ∈ I there is a j0 ∈ J, so that ij ≥ i0 for all j ≥ j0 . Note: A subnet of a sequence is not necessarily a subsequence. In a topological space (T, T ), we say that ...
... A subnet of a net (xi : i ∈ I) is a net (yj : j ∈ J), together with a map j → # ij from J to I, so that xij = yj for all j ∈ J for all i0 ∈ I there is a j0 ∈ J, so that ij ≥ i0 for all j ≥ j0 . Note: A subnet of a sequence is not necessarily a subsequence. In a topological space (T, T ), we say that ...
CONVERGENT SEQUENCES IN TOPOLOGICAL SPACES 1
... a 6= b. On the other hand, X is not Hausdorff since every two non-empty open sets have nontrivial intersection. There is still a broad class of topological spaces for which the converse of theorem 2.2 does hold true. To describe this class of spaces we first need the following definition. Definition ...
... a 6= b. On the other hand, X is not Hausdorff since every two non-empty open sets have nontrivial intersection. There is still a broad class of topological spaces for which the converse of theorem 2.2 does hold true. To describe this class of spaces we first need the following definition. Definition ...
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... Let X be a topological space, and C(X) the ring of continuous functions on X. A subspace A ⊆ X is said to be C-embedded (in X) if every function in C(A) can be extended to a function in C(X). More precisely, for every realvalued continuous function f : A → R, there is a real-valued continuous functi ...
... Let X be a topological space, and C(X) the ring of continuous functions on X. A subspace A ⊆ X is said to be C-embedded (in X) if every function in C(A) can be extended to a function in C(X). More precisely, for every realvalued continuous function f : A → R, there is a real-valued continuous functi ...
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... at x if there is a sequence (Bn )n∈N of open sets such that whenever U is an open set containing x, there is n ∈ N such that x ∈ Bn ⊆ U . The space X is said to be first countable if for every x ∈ X, X is first countable at x. Remark. Equivalently, one can take each Bn in the sequence to be open nei ...
... at x if there is a sequence (Bn )n∈N of open sets such that whenever U is an open set containing x, there is n ∈ N such that x ∈ Bn ⊆ U . The space X is said to be first countable if for every x ∈ X, X is first countable at x. Remark. Equivalently, one can take each Bn in the sequence to be open nei ...