Difference between Metric Space and Topological Space I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise convergence is NOT a metric concept." (page 222) I know the definitions for uniform and pointwise convergence of a sequence of functions. However, I could not understand the quote above. Could anyone explain it to me, please? Thank you!
Uniform convergence of a sequence of functions implies convergence of the sequence in the metric space of all functions.So it is a  metric concept.
Uniform convergence of a sequence of functions implies convergence of the sequence in the metric space of all functions.So it is a  metric concept.