Question 1:
Prove that the function max: R × R ---> R,   (x, y) ---> max{x,y} is continuous, where R × R have their Euclidean metrics.

Question 2:
Find a continuous bijection between topological spaces whose inverse is not continuous.

Question 3:
Let (X, T) be a topological space and take A ⊆ X.
Prove that there is a maximal open subset of X disjoint from A.
This is the exterior of A, denoted ext(A).

Question 4:
Let R[t] denote the set of all real polynomials in the intermediate t.
Let S be a subset of R[t].
Define V(S) := {x ∈ R | f(x) = 0 for every f ∈ S}.
Define F := {V(S) | S ⊆ R[t]}.
Prove that there is a topology on R, whose closed sets are precisely the elements of F and that this topology is not metrisable.
This is the Zariski topology on R and arises in algebraic geometry.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Topology Problems
    $25.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Topology Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats