Some forms of the closed graph theorem
... for all xe E, i.e. E is isomorphic to a subspace of c0. In (18), Webb introduces the notion of sequentially barrelled spaces; E is sequentially barrelled, if, whenever/„ ->- 0 cr(E', E) then (/„)£=x is equicontinuous. We remark that (I1, T(P, C0)) is sequentially barrelled (for it can be easily show ...
... for all xe E, i.e. E is isomorphic to a subspace of c0. In (18), Webb introduces the notion of sequentially barrelled spaces; E is sequentially barrelled, if, whenever/„ ->- 0 cr(E', E) then (/„)£=x is equicontinuous. We remark that (I1, T(P, C0)) is sequentially barrelled (for it can be easily show ...
1. The one point compactification Definition 1.1. A compactification
... 1. The one point compactification Definition 1.1. A compactification of a topological space X is a compact topological space Y containing X as a subspace. Given any non-compact space X, compactifications always exist. This section explores the smallest possible compactification obtained by adding a ...
... 1. The one point compactification Definition 1.1. A compactification of a topological space X is a compact topological space Y containing X as a subspace. Given any non-compact space X, compactifications always exist. This section explores the smallest possible compactification obtained by adding a ...
Connectedness in Isotonic Spaces
... to the discrete space {0, 1} is constant. The next result shows that we can extend this definition to isotonic spaces by assuming that {0, 1} is a T1 -isotonic space. Theorem 5.3 An isotonic space (X, cl) is connected if and only if for all T1 -isotonic doubleton spaces Y = {0, 1}, any continuous fu ...
... to the discrete space {0, 1} is constant. The next result shows that we can extend this definition to isotonic spaces by assuming that {0, 1} is a T1 -isotonic space. Theorem 5.3 An isotonic space (X, cl) is connected if and only if for all T1 -isotonic doubleton spaces Y = {0, 1}, any continuous fu ...
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... Summary. A manifold is a space that is locally like Rn , however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible loops, that distinguish it from the topologically trivial Rn . Standard Definition. An n-dimensional to ...
... Summary. A manifold is a space that is locally like Rn , however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible loops, that distinguish it from the topologically trivial Rn . Standard Definition. An n-dimensional to ...
Very dense subsets of a topological space.
... Definition (10.3.1). — A topological space X is Jacobson if the set of closed points X0 of X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in ...
... Definition (10.3.1). — A topological space X is Jacobson if the set of closed points X0 of X is very dense in X; that is, if X0 ,→ X is a quasi-homeomorphism. Proposition (10.3.2). — Let X be Jacobson, Z ⊆ X locally quasi-constructible. Then the subspace Z is Jacobson, and a point z ∈ Z is closed in ...