![On Topological Sets and Spaces - Global Journal of Science](http://s1.studyres.com/store/data/001873138_1-6ab443680b6a4228aa1b78fa4e1625c5-300x300.png)
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 ...
PDF
... 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 ...
PDF
... 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 ...