Advanced Calculus Problems

[Graystone]

1.) Let X and Y be sets and f:X→Y a function. Also let A с X and B с Y.

(a) A с f-1(f(A)) with equality when f is injective.

(b) f(f-1(B)) с B with equality when f is surjective.

 

2,) Let f:X→Y and g:Y→Z be functions.

(a) If g ◦ f is injective, then f is injective.

(b) If g ◦ f is surjective, then g is surjective.

 

3.) (a) Let R+ с R be the subspace of positive reals. Show that R is homeomorphic to R+ .

(b) Show that (a,b) is homeomorphic to (0,1) .

© Show that [a,b] is homeomorphic to [0,1] .

 

4.) Let X be a space.

(a) If X is Hausdorff, then every convergent sequence has a unique limit.

(b) X is Hausdorff if and only if the diagonal ΔX is closed in X×X .

 

5.) Let X be a metric space with metric d. Suppose that x є X and A is a closed subset not containing x . Show that there are disjoint open sets U and V containing x and A respectively.

 

 

I'm not looking for answers. I've never taken a proofs course, or linear algebra or anything of that sort. My math background is Cal3. I'm having a hard time understanding the terminology such as injective, surjective, homeomorphic, Hausdorf, etc. Can anyone help me understand these?

[Xittenn]

injective is a one to one mapping

 

surjective is a mapping in which all values in the set being mapped onto has a value associated with it i.e. if you have a set of apples and you have a corresponding set of tickets each apple will be represented by a ticket; there could be two tickets to an apple

 

homeomorphism is the mapping of a topological space with the properties bijective, is continuous in function, and is in continuous inverse function

 

A Hausdorf space requires a little more knowledge about mathematics to be properly understood; I don't properly understand a Hausdorff space

 

 

I usually avoid direct answers in homework but these definitions are readily available from wiki as a preliminary overview. I think the problem is you have picked up a book on intermediate/advanced topology and skipped set theory not so much proof theory although that would probably help as well. A more recent undergraduate text--like the ones you find in high school, graduate texts drop the bs and get straight to it, don't worry nobody is questioning your manliness--on topology would give you the details needed to proceed. I don't think it is proper to compare calculus to first order predicate calculus, it's not the same thing. Why it's not the same thing is a question I can't give proper give definition to, one is first order logic, the other is an academic treatment of continuous manifolds. First order logic is about building axioms and is not what people generally think of as calculus. One is the rules to defining a value of a problem, the other is rules to defining what a problem is.

Edited February 22, 2012 by Xittenn

[Graystone]

I'm starting to understand the terminology. I can easily write out the proof for number 2, but the wording on number 1 is severely tripping me up. The statement is just about self sufficient yet its part of the homework set. I may be mistaken but it reads to me as if it's saying "a chair is defined as a seat".

[DrRocket]

A Hausdorf space requires a little more knowledge about mathematics to be properly understood; I don't properly understand a Hausdorff space

 

A topological space is Hausdorff if given any two distinct point x andy there exist neighborhoods of x and y that are disjoint.

 

 

I think the problem is you have picked up a book on intermediate/advanced topology and skipped set theory not so much proof theory although that would probably help as well.

 

My guess is thatt he book is probably an introductory point set topology book. A book like that is not an "advanced calculus" text, and does not require calculus as a pre-requisite, but it does require a level of mathematical maturity and familiarity with constructing proofs that is beyond what one normally expects of a student in an advanced calculus class.

Edited February 23, 2012 by DrRocket