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  1. R

    Set of continuous bounded functions.

    Homework Statement This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year. If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...
  2. R

    Pointwise vs. Uniform Convergence.

    Homework Statement I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent. Homework Equations The relevant equation used is that lim (n-> infinity) sup|(fn(x) -...
  3. R

    Every sequence of bounded functions that is uniformly converent is uniformly bounded

    Homework Statement Prove that every sequence of bounded functions that is uniformly convergent is uniformly bounded. Homework Equations Let {fn} be the sequence of functions and it converges to f. Then for all n >= N, and all x, we have |fn -f| <= e (for all e >0). ---------- (1)...
  4. R

    Geometric interpretation of Generalized MVT

    Homework Statement I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation Homework Equations [f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x) The Attempt at a Solution On...
  5. R

    Simple question on continuity

    Homework Statement If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Homework Equations The Attempt at a Solution Since X is bounded in R, it is a subset of cell. And all cells in R are compact.All bounded sub...
  6. R

    Simple Question on continuity

    Homework Statement 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). Homework Equations The...
  7. R

    Simple question in topology (finite vs. infinite)

    Homework Statement I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable. 1)So, anything that is finitie is countable. Anything that is infinite is also countable? Then what is uncountable, something that...
  8. R

    Very badly stuck (Prove that X is compact)

    Homework Statement If X is a metric space such that every infinite subset has a limit point, then prove that X is compact. Homework Equations Hint from Rudin: X is separable and has a countable base. So, it has countable subcover {Gn} , n=1,2,3..... Now, assume that no finite sub...
  9. R

    Please comment. (Topology question on separability)

    Homework Statement If X be a metric space in which every infinite subset has a limit point, then X is separable. This is a question from Rudin but I am having some difficulty just understanding how to use the hint. Homework Equations The hint as in the book is . Fix delta >0, and...
  10. R

    A simple problem in Real Analysis

    Homework Statement I am having somewhat a difficult time just understanding a simple concept. I am trying to prove that every open subset G of a separable metric space X is the union of a sub collection {Vi} such that for all x belongs to G, x belongs to some Vi (subset of G). I am...
  11. R

    A metric space having a countable dense subset has a countable base.

    1. Homework Statement Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi. 2. Homework Equations Hint from the book of Rudin: Center the point...
  12. R

    Every separable Metric space has countable base.

    Homework Statement Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi. Homework Equations Hint from the book of Rudin: Center the point in a...
  13. R

    Prove that the closure is the following set.

    Homework Statement Suppose (X,d) is a metric space. For a point in X and a non empty set S (as a subset of X), define d(p,S) = inf({(d(p,x):x belongs to S}). Prove that the closure of S is equal to the set {p belongs to S : d(p,S) =0} Homework Equations Closure of S = S U S' , where S'...
  14. R

    Every open set contains both open and closed cell.

    Homework Statement If an open cell is defined as (a1,b1) X (a2,b2) X .... (an,bn) in R^n and closed cell is defined as [a1,b1] X [a2,b2] X .... [an,bn], then every open set in R^n contains an open-n-cell and a closed-n-cell. Homework Equations Def: An open set is a set which has all...
  15. R

    Prove that a set is open.

    Homework Statement In general, in R^n, what is the best way to approach the problem - a given set is open? The given set E is such that for all x,y that belong to the given set, d(x,y) < r. Homework Equations The Attempt at a Solution let x be the center of the sphere and y be...
  16. R

    Open n-cell is open?

    Homework Statement An open-n-cell in R^n defined as (a1,b1) x (a2,b2) x ....... (an, bn). Prove that every open n-cell is open. Homework Equations The Attempt at a Solution I was thinking of using induction. Clearly the base case n=1 is open as (a1,b1) is open in R1. It is a...
  17. R

    Is every point of every closed set E subset of R^2 a limit point of E?

    Homework Statement If E is subset of R^2, then is every point of every closed set E, a limit point of E? Homework Equations The Attempt at a Solution I think the answer is yes. Consider E = { (x,y) | x^2 + y^2 <= r^2} , where r is the radius. Consider a point p that belongs...
  18. R

    Is (a,b) open in R^n

    Homework Statement We know that (a,b) is open in R. But is it open R^n? Homework Equations The Attempt at a Solution I don't think (a,b) is open in R^n even if it is open in R. Let's take for example n=2, then E = {(x,y) | x^2+y^2 < r^2} , where r^2 = |b|^2 + Y^2 , for all...
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