Recent content by Ka Yan

Could someone kind enough to tell me, please?

Is my reasoning correct? Original problem: In the finite completement topology on R(denoted by FCTR here), to what point or poionts does the sequense x[SIZE="1"]n = 1/n converge? I firstly prove that R with FCT does not a Hausdorff. Let T[SIZE="1"]f be FCTR, x[SIZE="1"]1, x[SIZE="1"]2...

What's the difference between those assertions: " A set X is both open and closed." and " A set X is neither open nor closed." For the first, I knew some examples: The real line itself, and the empty set. But what example can be araised about the second? And any better ones to the...

Homework Statement Let X be an ordered set. If Y is a proper subset of X that is convex in X, does it follow that Y is an interval or a ray in X? The Attempt at a Solution I considered it to be yes. Since in the ordinary situation, the assertion is obviously valid: check out the...

Why is that the Banach-Tarski Paradox (Theorem, say) ture for R^n of n\geq3, but nor for R and R^2, please?

I overheard there were some paradox which brought troubles of AC and even Set Theory to mathematicians, I wonder what was them, what troubles did they lead to, have mathematicians solved them already, how, or why not?

I, by the first time, came across with the Axiom of Choice today, found it beautiful, of course. And I'm interested in seeing, ladies and gentlemen here, as mathematicians, what are your attitude towards that axiom, I mean, the stronger one (i.e. the infinite axiom of choice). For example, do...

Yes, thanks gentlemen. That, I was originally considered, if f is a function on a subset B of A, then an "extension" of f from B to A is that, exist a function g on A, such that g(x)=f(x) for all x in B. But B and A are seem necessary to be closed, and I didnt quite sure if those A and B...

There is a simple problem, and I gave my simple prove. Could anybody help me check whether it is correct: Show that if B is not finite and {B}\subset{A}, then A is not finite. My prove: Since B is not finite, there exists a bijection of B into one of its proper subset, C, say, and denote...

Two questions need helps I got two questions below need helps: 1. Let f be a real continuous function defined on a closed subset E of R^1, then how can I prove the existence of some corressponding real continuous functions g on R^1, such that g(x)=f(x) for all x\inE ? 2. Let f and g two...

Could anybody help me check whether my judgements ture or false? (MJ = My Judgement) Suppose f maps A into B, and g maps B into C 1. If f and g are injective, then g\circf is injective; (MJ)but that when g\circ f is injective, the injectivity of f and g are unsure. 2. If f and g are...

Thanks a lot, gentlemen! Erm, may I BTW ask one more away-from-the-point question here? That, could anybody, if possible, provide me of any forums where Economics be discussed? Since I have just started the journey of self-learning of the Micro and Macroeconomics, but found questions nowhere...

If f is a real function on R^1, and holds:lim [f(x+h)-f(x-h)] = 0 for every x belongs R^1. Does f continuous? And I thought it no. Since I considered it mentioned only the left-hand and right-hand limit are equal, but whether or not equal to f(x) was not exactly known. Will anybody provide...

But mister, I wonder if I can apply the mean by L'Hospital's Law. I made up that example, because I worked it up with the L. Law, and found it goes to 0, but I didn't quite sure that if an can satisfy the condictions so that the law works.

In fact, for Q1, as anothter example, I can just simply let Sn = ln(n+1), and there I will get it.