Construction of Spaces
... Def. Let i : X Y be an inclusion and Y have topology S . Then, ( X , i * S ) is said to have the subspace topology. Prop. If W is a subspace of X and X is a subspace of Y , then W is a subspace of Y . Proof. Let j : W X and i : X Y be inclusions . (i j ) 1 (V ) j 1 (i 1 (V )) for all op ...
... Def. Let i : X Y be an inclusion and Y have topology S . Then, ( X , i * S ) is said to have the subspace topology. Prop. If W is a subspace of X and X is a subspace of Y , then W is a subspace of Y . Proof. Let j : W X and i : X Y be inclusions . (i j ) 1 (V ) j 1 (i 1 (V )) for all op ...
June 2010
... Do three of the problems from section A and three questions from section B. If you work more than the required number of problems, make sure that you clearly mark which problems you want to have counted. If you have doubts about the wording of a problem or about what results may be assumed without p ...
... Do three of the problems from section A and three questions from section B. If you work more than the required number of problems, make sure that you clearly mark which problems you want to have counted. If you have doubts about the wording of a problem or about what results may be assumed without p ...
Topology (Maths 353)
... (1) f ∗ OY is a topology on X, called the induced topology (more precisely, the topology induced by f ); (2) f : (X, f ∗ OY ) → (Y, OY ) is continuous; (3) f ∗ OY is the smallest topology on X with this property (i.e., every topology w.r.t. which the map f is continuous contains f ∗ OY ). Theorem. L ...
... (1) f ∗ OY is a topology on X, called the induced topology (more precisely, the topology induced by f ); (2) f : (X, f ∗ OY ) → (Y, OY ) is continuous; (3) f ∗ OY is the smallest topology on X with this property (i.e., every topology w.r.t. which the map f is continuous contains f ∗ OY ). Theorem. L ...
MT 3803 - Loyola College
... (ii) Let X be a non–empty set, and let there be give a “closure” operation which assigns to each subset A of X a subset A of X in such a manner that (1) = , (2) A A , (3) A A , and (4) A B = A B . If a “closed” set A is defined to be one for which A = A , show that the class of all compl ...
... (ii) Let X be a non–empty set, and let there be give a “closure” operation which assigns to each subset A of X a subset A of X in such a manner that (1) = , (2) A A , (3) A A , and (4) A B = A B . If a “closed” set A is defined to be one for which A = A , show that the class of all compl ...
Theorem 2.1. Tv is a topologyfor v. Definition. For each x EX, 1T(X)is
... Suppose (X, T) is a topological space, and suppose V is a collection of mutually exclusive subsets of X whose union is X (i.e. a partition of X). We wish to define a new topological space whose points are the members of V. To that end we define a topology Tv on V as follows: a subset F of V belongs ...
... Suppose (X, T) is a topological space, and suppose V is a collection of mutually exclusive subsets of X whose union is X (i.e. a partition of X). We wish to define a new topological space whose points are the members of V. To that end we define a topology Tv on V as follows: a subset F of V belongs ...
Contents 1. Topological Space 1 2. Subspace 2 3. Continuous
... Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f −1 (V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such that f (U ) ⊂ V. Proposition 3.1. ...
... Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f −1 (V ) is open in X whenever V is open in Y. A function f : X → Y is called continuous at x if for every open neighborhood of f (x), there exists an open neighborhood U of x such that f (U ) ⊂ V. Proposition 3.1. ...
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.