I did really enjoy geometry in college. It's been a while since I've thought about any topology, but I can give it a shot, if you've got a specific question.
So let there be the metric space (X, d), let C(N, R) be the set of all convergent sequences in R and let there be the function f: C(N, R) |--> X; f(x_n) --> lim x_n. There's two situations: one where the distance associated with C(N, R) is d_infinity, which is the maximum difference for all elements in the two sequences in C(N, R), and one where the distance associated with C(N, R) is d(x, y) = 1/n where n is the smallest k where x_k != y_k. I need to prove that f is continuous in the first situation and not continuous in the second situation.
Imma make a glossary in case someone wants to help but doesn't know the terminology.
Metric space: it's a set associated with a function d that fulfills the properties of non-negativity, commutativity and triangular inequality. The function d is called the distance.
Distance: it's the function d associated with a metric space.
Sequence: an enumerated collection of objects. Like a set, but order matters. The elements of a sequence x_n are denominated by their order x_1, x_2, x_3, and so on
Convergence: a sequence is said to be convergent when it has a limit.
Limit of a sequence: a is the limit of a sequence when for every epsilon > 0 there's a natural number n0 such that for every n > n0 you have |x_n - a| < epsilon. If a sequence doesn't have a limit, it's called a divergent sequence.
Continuity of a function in metric spaces: a function f: X |--> Y is said to be continuous when for every epsilon > 0 there's a delta > 0 such that d1(x, y) < delta implies d2(f(x), f(y)) < epsilon. In this example, x and y are any random elements of X and d1 and d2 are the distances associated with the metric spaces X and Y.
Yeah, probably. I am not the best person to look for help in analysis.
I never felt I had a grasp on what was going on there. I really need to revisit it now that I have a more mature view of mathematics.
Dammit, you probably can't help me then. Any chance you're good with metric spaces? It's what I really need right now
I did really enjoy geometry in college. It's been a while since I've thought about any topology, but I can give it a shot, if you've got a specific question.
So let there be the metric space (X, d), let C(N, R) be the set of all convergent sequences in R and let there be the function f: C(N, R) |--> X; f(x_n) --> lim x_n. There's two situations: one where the distance associated with C(N, R) is d_infinity, which is the maximum difference for all elements in the two sequences in C(N, R), and one where the distance associated with C(N, R) is d(x, y) = 1/n where n is the smallest k where x_k != y_k. I need to prove that f is continuous in the first situation and not continuous in the second situation.
Imma make a glossary in case someone wants to help but doesn't know the terminology.
Metric space: it's a set associated with a function d that fulfills the properties of non-negativity, commutativity and triangular inequality. The function d is called the distance.
Distance: it's the function d associated with a metric space.
Sequence: an enumerated collection of objects. Like a set, but order matters. The elements of a sequence x_n are denominated by their order x_1, x_2, x_3, and so on
Convergence: a sequence is said to be convergent when it has a limit.
Limit of a sequence: a is the limit of a sequence when for every epsilon > 0 there's a natural number n0 such that for every n > n0 you have |x_n - a| < epsilon. If a sequence doesn't have a limit, it's called a divergent sequence.
Continuity of a function in metric spaces: a function f: X |--> Y is said to be continuous when for every epsilon > 0 there's a delta > 0 such that d1(x, y) < delta implies d2(f(x), f(y)) < epsilon. In this example, x and y are any random elements of X and d1 and d2 are the distances associated with the metric spaces X and Y.