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Homework Statement Define the interior A◦ and the closure A¯ of a subset of X. Show that x ∈ A◦ if and only if there exists ε > 0 such that B(x,ε) ⊂ A. The Attempt at a Solution [/B]

Here is my new solution, but I'm not sure if the graph I put is of any use, but if you can, please put up a similar graph to the one I put to help better understand.

aah, I think I get it now. Thanks for the reply

Right now, I am studying Advanced Calculus (proof based), and it is hard thinking through some of the definitions without first thinking about it concretely (meaning how to visualize it better geometrically, if that makes any sense?). I need help with this definition. Definition Let X be a...

Classical or modern physics? Basically, which branch is responsible for all modern inventions: Cars, airplanes, modern sea ships, spacecraft s, microwaves, ovens, computers, televisions, smartphones, etc. (Also, how did we get to the moon, using concepts from classical or modern physics?)...

When I asked this question in another forum, some people argued that scientific and mathematical discoveries are born out of curiosity and a way to explain unknown elements that surrounds us. Others say it comes from originality. In my opinion, I think it does come from curiosity, but you have...

I don't know why I think purer math is more important for CS majors while applied math is more important for Physics majors. Only because for CS, you need to be able to do logic and think logically and be able to do proofs (but then again, I'm not a CS major, nor have I taken discrete math, but...

What about for computer scientists? As far as I have seen in a few colleges with different requirements, it seems that discrete math is the only required math course. Some colleges I've looked at also require students to take Linear Algebra (especially those specializing in graphics) and two...

As I stated before, I'd imagine that computer scientists know more of the abstract kinds of math that deal with logic and algorithms while physicists know more of the applied math topics that apply to everyday phenomenon. Those pure math courses (real analysis, abstract algebra, topology) seem...

I am wondering, what is the math level of your average CS or physics major? Like how much math do these two take beyond stuff like the Calculus sequence, differential equations and linear algebra? I was having a discussion with one of my old math professors from my community college (I'm in a...

Didn't know where to put this question. but just wanted to ask quickly. All i know is that in Calculus 3, you use double, triple or multiple integrals for 3D models. But what other higher level math course uses or involves multiple integrals?

That is what I'm trying to point out. Along with an advisors note of my promises, I want to show them my plan for the next year as well to support my appeal.

How would it not be helpful? At least I am letting them know my plan for the upcoming Junior Year? Do I have to tell them the hours I plan on studying as well. What about telling them that I will take advantage of the resources given to me such as tutors, going to professors office for help, etc?

But should I provide my plan for the following Junior year, like the classes I plan on taking for both the fall and spring semester. Should I let them know I am re-studying my Calculus and Chemistry notes for when I am taking the next classes?

What do I say to make them believe? Should I say by turning off my house computer and only studying at the library? Being at home is my comfort zone and can be very distracting.

I plan on taking three classes over the next fall and spring semester. My plan would be to at least study around 4-6 hours for each class. Over the shorter session, I plan on taking other courses I have interest in like history. Would that be a good plan?

As for additional documentation, should I list down the courses I plan to take for the next fall and spring semester.

I re-edited my letter Dear Academic Standing Committee, My name is xxxxx and I am majoring in Liberal Arts: Math and Science. I am writing this letter to appeal my academic dismissal due to failure of maintaining an adequate GPA and inadequate grades in several courses. There is absolutely no...

The longer session is the normal 12 week semester (September to December in the fall or March to June in the spring). The shorter session take place over a 6 week semester (January to February in the winter or June to July in the summer). In my college, the fall semester is divided into session...

So how do I make my plan seem strong? Can I say something like: I plan on volunteering or working somewhere to help me concentrate better? And I have thought of that for a while now. Part of the reason for having a poor GPA and poor grades in several courses was due to being irresponsible and...

I started a previous thread regarding my academic appeal, but was not able to access it due to internet issues over the past two days, so I am starting a new one. <Moderator's note: post has now been added to the existing thread.> I rewrote my academic appeal and got some advise from different...

I revised it again Dear Academic Standing Committee, My name is xxxxxxx and I am writing this letter to appeal my academic dismissal due to failure of maintaining an adequate GPA and not performing well in classes and to apologize for not doing well as I am supposed to for college. I took...

How does this sound? I re-edited it: Dear Academic Standing Committee, My name is xxxxx and I am writing this letter to appeal my academic dismissal due to failure of maintaining an adequate GPA and not performing well in classes. The reason behind my poor academic performance was due to poor...

How about this? Is there something wrong with my letter? Did it have a lot of grammatical error?

Why, does it sound bad? Does it have a lot of grammatical error? I'm trying to write a statement about why I was doing bad in the first place and I want to tell them that I won't screw up this time.

How do I beg? That will only make me look bad. How do I apologize without first explaining why I never maintained a good standing GPA in the first place. How do I cut down my letter so I don't make it too long. By the way, it's a community college. Yes, it seems ridiculous why I would do bad in...

Please help. I am not confident about my academic suspension appeal. Please help me with my grammar, spelling, punctuation. And if there is something wrong with my letter, please address it but don't be rude about it. I am worried someone might criticize my letter and start making judgements, so...

I got a D in Gen Chem 1 and a D+ in Gen Chem 2. Mostly because of missing homeworks and I was taking 5 classes at the same time. But when I restudied both of them, and took Organic Chemistry, I managed to get A- in both Orgo 1 and 2. How will graduate school see this? And also, if I were to...

Thank you. Just curious, what about at your university? Do they allow you to take them at the same time IF you took both already and passed them?

I have already taken both General Chemistry 1 and 2 at my community colleges. I did not get a good grade, but that was because of I missed a lot of homeworks and projects, but did good on the tests and I understand some of the material that I learned. In general, can you take both of them again...

Please explain Nuclear Magnetic resonance and how it is applied to other fields? I know it has to do with magnetic field around atoms and stuff like spectroscopy and imaging. But please correct me and please go in dept with Nuclear magnetic resonance.

Does the Physics sequence go something like this: For Physics 1, you will need Calculus 1 and for Physics 2, you will need Calculus 2.

I wasn't assigned a textbook yet

What calculus method do I need to apply this question?

I am about to take General Physics 1 (Calculus version) in the spring and I am nervous. What is it like? Can you post an example of the kind of questions I will be given in Physics 1.

And are they optional?

Thank you, but as for math, after those courses, in general for all colleges/universities, what goes after calculus, LA and DQ

I know that after completing general chemistry, there is organic chemistry and physical and biochemistry, but what about for physics and biology? As for those math courses after calculus, linear algebra and differential equations, what goes after that? It's not like I'm going to every upper...

I mean in general, what goes after those intro classes. What do i take after i completed biology 1 and 2 and chemistry and physics? I go to a community college and it's kind of limited. Only intro courses and that's it.

I am a major in liberal arts and science ( or at least still trying to decide on which STEM field i will be going in ) and right now i am going to take intro science courses (general biology 1 and 2, chemistry 1 and 2, physics 1 and 2) and also math courses. After i am done with the intro...