Final exam questions
... boundary of A in X, Int(A) denote the interior of A, and A0 denotes the set of limit points of A in X. Important note: Problems 30-49 must be done before you start working through problems 1-29. Exercises adapted from Introduction to Topology by Baker. 1. If f : X → Y is a function and U and V are s ...
... boundary of A in X, Int(A) denote the interior of A, and A0 denotes the set of limit points of A in X. Important note: Problems 30-49 must be done before you start working through problems 1-29. Exercises adapted from Introduction to Topology by Baker. 1. If f : X → Y is a function and U and V are s ...
Fixed Point in Minimal Spaces
... Definition 11. (X, M) is said to have the fixed point property if every m-continuous function f : X → X has a fixed point. Example 2. Suppose X = {x1 , x2 , x3 } and M = {∅, {x1 }, {x2 }, X} is a minimal structure on X. In order to show that X has the fixed point property it is enough to show that a ...
... Definition 11. (X, M) is said to have the fixed point property if every m-continuous function f : X → X has a fixed point. Example 2. Suppose X = {x1 , x2 , x3 } and M = {∅, {x1 }, {x2 }, X} is a minimal structure on X. In order to show that X has the fixed point property it is enough to show that a ...
Distance and Isometries Reading Part 1
... • Theorem (Corresponding Angle Theorem for absolute geometry): In absolute geometry, given two lines cut by a transversal, if corresponding angles are congruent, then the two lines are parallel. There is an important recurring question in geometry. We could call it THE BIG QUESTION: • Given a line L ...
... • Theorem (Corresponding Angle Theorem for absolute geometry): In absolute geometry, given two lines cut by a transversal, if corresponding angles are congruent, then the two lines are parallel. There is an important recurring question in geometry. We could call it THE BIG QUESTION: • Given a line L ...
Reflecting properties in continuous images of small weight
... Theorem 3.1 A pseudocompact Tychonoff space X has the property that every continuous image Y of X with w(Y ) ω1 has countable pseudocharacter if and only if X is compact and perfect. Proof To simplify the terminology, let us say that a space X has property P if it is a pseudocompact Tychonoff spac ...
... Theorem 3.1 A pseudocompact Tychonoff space X has the property that every continuous image Y of X with w(Y ) ω1 has countable pseudocharacter if and only if X is compact and perfect. Proof To simplify the terminology, let us say that a space X has property P if it is a pseudocompact Tychonoff spac ...
