Thm 27.1: Let X by a simply ordered set having the least upper
... Thm 27.4 (Extreme value thm). f cont : (X, compact) → (Y, ordered) implies there exists c, d ∈ X such that f (c) ≤ f (x) ≤ f (d) for all x ∈ X. Idea of proof: If f (X) has no largest element, then f (X) ⊂ ∪y∈f (X) (−∞, y) ...
... Thm 27.4 (Extreme value thm). f cont : (X, compact) → (Y, ordered) implies there exists c, d ∈ X such that f (c) ≤ f (x) ≤ f (d) for all x ∈ X. Idea of proof: If f (X) has no largest element, then f (X) ⊂ ∪y∈f (X) (−∞, y) ...
POINT SET TOPOLOGY Definition 1 A topological structure on a set
... Theorem 18 A necessary and sufficient condition that F 0 with base B 0 be finer than F with base B id B 0 ⊂ B. Examples (i) Let X be a non-empty partially ordered set (≤) in which each pair of elements has an upper bound. The sections {x : x ≥ a} of X form a filter base. The filter it defines is cal ...
... Theorem 18 A necessary and sufficient condition that F 0 with base B 0 be finer than F with base B id B 0 ⊂ B. Examples (i) Let X be a non-empty partially ordered set (≤) in which each pair of elements has an upper bound. The sections {x : x ≥ a} of X form a filter base. The filter it defines is cal ...
COUNTABLE PRODUCTS 1. The Cantor Set Let us constract a very
... fn (x), but fn (a) ∈ / [0, 2 ), for all a ∈ A. This implies fn (A) ⊆ [ 2 , 1]. As the set [ 2 , 1] is / fn (A) and thus the family {fi : i ∈ N} closed, this implies fn (A) ⊆ [ 2 , 1]. Hance fn (x) ∈ separates points and closed spaces. ...
... fn (x), but fn (a) ∈ / [0, 2 ), for all a ∈ A. This implies fn (A) ⊆ [ 2 , 1]. As the set [ 2 , 1] is / fn (A) and thus the family {fi : i ∈ N} closed, this implies fn (A) ⊆ [ 2 , 1]. Hance fn (x) ∈ separates points and closed spaces. ...
Internet Topology
... Web of interconnected networks Grows with no central authority Autonomous Systems optimize local communication efficiency The building blocks are engineered and studied in depth Global entity has not been characterized ...
... Web of interconnected networks Grows with no central authority Autonomous Systems optimize local communication efficiency The building blocks are engineered and studied in depth Global entity has not been characterized ...
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.