Please answer e and f only 2-5. For each of the following properties, give an example consisting of two sub-sets X, Y CR2, both considered as topological spaces with their Euclideantopologies, together with a map f: X → Y that has the indicated property.(a) f is open but neither closed nor continuous.(b) f is closed but neither open nor continuous.(c) f is continuous but neither open nor closed.(d) f is continuous and open but not closed.(e) f is continuous and closed but not open.(f) f is open and closed but not continuous.

Answered: Image /qna-images/answer/8c0abb34-3368-4f1a-bcc0-ae8d468cdd8f.jpg

Answered: (e) f is continuous and closed but not…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.3: Properties Of Composite Mappings (optional)

Problem 3TFE: Label each of the following statements as either true or false. 3. Let , , and be mappings from ...

See similar textbooks

Similar questions

Label each of the following statements as either true or false. 3. Let , , and be mappings from into such that . Then .
Label each of the following statements as either true or false. 4. Let , , and be mappings from into such that . Then .
Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.
Label each of the following statements as either true or false. 9. Composition of mappings is an associative operation.
2-5. For each of the following properties, give an example consisting of two sub- sets X, Y CR2, both considered as topological spaces with their Euclidean topologies, together with a map f: X → Y that has the indicated property. (a) f is open but neither closed nor continuous. (b) f is closed but neither open nor continuous. (c) f is continuous but neither open nor closed. (d) f is continuous and open but not closed. (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.
Define what it means for a topological space to be completely regular,
Could you explain to me how to show 4.2 in detail?
How do you solve this question on discrete mathematics: 2. Let A be the set of all points in three-dimensional space. Define R as the relation that two points will have the same z coordinate. (a) Give three examples that are related to P(4,6,2). (b) Prove or disprove that R is reflexive. (c) Prove or disprove that R is symmetric. (d) Prove or disprove that R is transitive. (e) Is R an equivalence relation? (f) What space do all points related to each other (e.g. all points withz= 2) describe?
2.5 For each of the following properties, give an example consisting of two subsets X, Y CR², both considered as topological spaces with their Euclidean topologies, together with a map f : X Y that has the indicated property. (a) f is open but neither closed nor continuous. (b) f is closed but neither open nor continmuous. (c) f is continuous but neither open nor closed. (d) f is contimuous and open but not closed. (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.
Let f : X → Y and g: Y → Z be two maps. (a) Show that if g o f is surjective, then g must be surjective. (b) Show that if g o f is injective, then f must be injective.
7-10. Let X and Y be topological spaces. Show that if either X or Y is con- tractible, then every continuous map from X to Y is homotopic to a constant map.
Let (¥,r) and (¥ ,4) be two topological spaces. Prove that a function f :(X .r)——(¥ ,A) is continuous if and only if the inverse image under / of every A- closed set is a r- closed set.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

(e) f is continuous and closed but not open. (f) f is open and closed but not continuous.