Number of connected components and continuous maps, Topology
Not what you're looking for?
The following question is answered:
Let f:X-->Y be a continuous onto map. Let D be a subset of Y such that YD has at least n connected components. Prove that Xf^(-1)(D) has at least n connected components.
Are the line and the plane with their usual topology homeomorphic?
Purchase this Solution
Solution Summary
It is proved that the number of connected components of the inverse image of a set by a continuous onto map can not decrease. We use the result to answer the question of whether a line and a plane with their usual topologies are homeomorphic. The usual topology homeomorphic is examined.
Solution Preview
Please see attachment for properly formatted copy.
Let A_1,...,A_n be disconnected components of YD. We are done if we show that f^{-1}(A_1),...,f^{-1}(A_n) are
disconnected in X. Indeed, as you can verify easily
f^{-1}(YD)=f^{-1}(Y) f^{-1}(D)=X f^{-1}(D),
since f^{-1}(Y)=X. It follows that since A_i is a subset of YD, then f^{-1}(A_i) is a subset of Xf^{-1}(D), and then under our assumption, f^{-1}(A_1),...,f^{-1}(A_n) are disconnected in X and so disconnected ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Probability Quiz
Some questions on probability