Topology : Past Exam Paper
Not what you're looking for?
Please see the attached file for the fully formatted problems.
B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant.
(b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism.
(c) (i) Recall that Fr E, the frontier of a subset E of a topological space X isdefined as Efl(X?E). Prove that E=EUFrE.
(ii) A topological space is 0 dimensiona1 if and only if whenever x C
V and V is open, there is an open set U with empty frontier such
that x e U C V. Show that the rationals in the relative topology as
a subset of JR with the usual topology is a 0-dimensional set.
(iii) Show that being 0-dimensional is a topological invariant.
Purchase this Solution
Solution Summary
Homomorphisms are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
QB5
a. A homomorphism between topological space and is a continuous bijection such that is also continuous. A topological invariant of a space is a property that depends only on the topology of the space, i.e. it is shared by any topological space homeomorphic to .
b. Proof:
is open means that for any open ...
Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Probability Quiz
Some questions on probability
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.