LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
... topological space X . Show that C ( X , ¡ ) is a real Banach space with respect to pointuise addition and scalar multiplication and the norm defined by f = sup f ( x) ; (2) if multiplication is defined pointuise, C ( X , ¡ ) is a commutative real algebra with ...
... topological space X . Show that C ( X , ¡ ) is a real Banach space with respect to pointuise addition and scalar multiplication and the norm defined by f = sup f ( x) ; (2) if multiplication is defined pointuise, C ( X , ¡ ) is a commutative real algebra with ...
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... The meaning of the word neighborhood in topology is not well standardized. For most authors, a neighborhood of a point x in a topological space X is an open subset U of X which contains x. If X is a metric space, then an open ball around x is one example of a neighborhood. Unless otherwise specified ...
... The meaning of the word neighborhood in topology is not well standardized. For most authors, a neighborhood of a point x in a topological space X is an open subset U of X which contains x. If X is a metric space, then an open ball around x is one example of a neighborhood. Unless otherwise specified ...
... Definition 2.3[3]: A topological space (X, τ) is said to be g*-additive if arbitrary union of g*closed sets is g*-closed. Equivalently arbitrary intersection ofg*-open sets is g*-open. Definition 2.4[3]: A topological space (X, τ) is said to be g*-multiplicative if arbitrary intersection of g*-close ...
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... 0 or 1, and whichever distance is smaller will be the upper bound on the radius of our ball. Since we can do this for any number in the interval, it must be open. On the other hand, an example of a closed set would be any closed interval. If we now consider the closed interval [0, 1], in order to ve ...
... 0 or 1, and whichever distance is smaller will be the upper bound on the radius of our ball. Since we can do this for any number in the interval, it must be open. On the other hand, an example of a closed set would be any closed interval. If we now consider the closed interval [0, 1], in order to ve ...
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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 ...
Padic Homotopy Theory
... p-adic homotopy theory There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is ...
... p-adic homotopy theory There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.