help with 2 problems about compact/pointwise convergence from Munkres - Topology

by Hells_Kitchen
Tags: compact or pointwise, convergence, munkres, topology
Hells_Kitchen is offline
Dec7-09, 10:04 PM
P: 62
Hi everyone,

I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are:

1. Consider the sequence of continuous functions fn : ℝ -> ℝ defined by fn(x) = x/n . In which of the following three topologies does this sequence converge: uniform, compact convergence, pointwise convergence? Answer the same question for the sequence given as:

fn(x)= 1 / [n^3 * (x - 1/n)^2 + 1]

2. Let (Y,d) be a metric space; let fn: X -> Y be a sequence of continuous functions; let f: X -> Y be a function (not necessarily continuous). Suppose that fn converges to f in the topology of pointwise convergence. Show that if {fn} is equicontinuous then f is continuous and fn converges to f in the topology of compact convergence.
Phys.Org News Partner Science news on
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered

Register to reply

Related Discussions
can you please help me proving , uniform convergence implies pointwise convergence Calculus & Beyond Homework 1
Munkres' Topology Science & Math Textbook Listings 2
Question about pointwise convergence vs. uniform convergence Calculus 2
Set theory in Munkres Topology General Math 3
topology by munkres Science & Math Textbook Listings 3