Chapter 1: Topology
... A topological space is a set S together with a collection O of subsets called open sets such that the following are true: i) the empty set ∅ and S are open, ∅, S ∈ O ii) the intersection of a finite number of open sets is open; if U1 , U2 ∈ O, then U1 ∩ U2 ∈ O iii) S the union of any number of open ...
... A topological space is a set S together with a collection O of subsets called open sets such that the following are true: i) the empty set ∅ and S are open, ∅, S ∈ O ii) the intersection of a finite number of open sets is open; if U1 , U2 ∈ O, then U1 ∩ U2 ∈ O iii) S the union of any number of open ...
Lecture 1
... 1.1.3 Example. Let X be the set consisting of just one point. Such a set has a unique topology consisting of all the subsets of X. This topological space is denoted by ∆0 or D0 or R0 and is called the point. 1.1.4 Definition. Let X and Y be topological spaces. A function f : X −→ Y is called contin ...
... 1.1.3 Example. Let X be the set consisting of just one point. Such a set has a unique topology consisting of all the subsets of X. This topological space is denoted by ∆0 or D0 or R0 and is called the point. 1.1.4 Definition. Let X and Y be topological spaces. A function f : X −→ Y is called contin ...
Problems
... 1. Write function Xn = mspolygon(X,x0,a) that scales the INPUT polygon by a (a>0) and moves its center to point x0, and draws both polygons in one image. The polygon is given by matrix X whose columns are the nodes (corner points) of the polygon. The output Xn is the nodes of new polygon. Define ...
... 1. Write function Xn = mspolygon(X,x0,a) that scales the INPUT polygon by a (a>0) and moves its center to point x0, and draws both polygons in one image. The polygon is given by matrix X whose columns are the nodes (corner points) of the polygon. The output Xn is the nodes of new polygon. Define ...
§4 谓词演算的性质
... if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-toone. The definition of one to one may be restated in the ...
... if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-toone. The definition of one to one may be restated in the ...
Lecture 3: Jan 17, 2017 3.1 Topological Space in Point Set Topology
... x-axis. For simplicity, assume f : X → Y, where X = Y = R. To be continuous, the pre-image in X of every open set in Y must be open. Consider an open set (−, ) ∈ Y. Its pre-image is the set of all points x ∈ X such that f (x) is in the open set (−, ). Look at any such open set in Y where || > 1 ...
... x-axis. For simplicity, assume f : X → Y, where X = Y = R. To be continuous, the pre-image in X of every open set in Y must be open. Consider an open set (−, ) ∈ Y. Its pre-image is the set of all points x ∈ X such that f (x) is in the open set (−, ). Look at any such open set in Y where || > 1 ...