Munkres Topology - Chapter 7 - Complete Metric Spaces and Function Spaces

In summary, the conversation discusses whether it is possible or advisable to read Chapter 7 of Munkres' "Complete Metric Spaces and Function Spaces" without prior knowledge of Tietze Extension Theorem, Imbeddings of Manifolds, and the entirety of Chapters 5 and 6. The suggestion is to start reading and looking up unfamiliar terms in the index as needed.
  • #1
sammycaps
91
0
Hello, I was wondering if it was possible (or advisable) to read Chapter 7 of Munkres (Complete Metric Spaces and Function Spaces) without having done Tietze Extension Theorem, the Imbeddings of Manifolds section, the entirety of Chapter 5 (Tychonoff Theorem) and the entirety of Chapter 6 (Metrization Theorems and Paracompactness)? I've done everything through The Urysohn Metrization Theorem (which is nearly the end of Chapter 4, right before Tietze Extension Theorem).

I expect the exercises might have some stuff from these areas, but flipping through the text quickly it *seems* I should be alright. But, I don't know. Any suggestions are welcome.

Thanks!
 
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  • #2
I suggest you just start doing it, and if you come across some terms you don't understand, look them up in the index, get a handle on them, then return to where you were. You may have to recurse a few levels. That's how everybody reads math books anyway.
 
  • #3
Tinyboss said:
I suggest you just start doing it, and if you come across some terms you don't understand, look them up in the index, get a handle on them, then return to where you were. You may have to recurse a few levels. That's how everybody reads math books anyway.

Alright, thanks!
 

FAQ: Munkres Topology - Chapter 7 - Complete Metric Spaces and Function Spaces

1. What is a complete metric space?

A complete metric space is a type of mathematical space that is equipped with a metric, which is a function that measures the distance between any two points in the space. A complete metric space is one in which every Cauchy sequence (a sequence where the terms get arbitrarily close to each other) converges to a point in the space.

2. How is completeness related to compactness in metric spaces?

Completeness and compactness are both important properties of metric spaces. A metric space is compact if it is both complete and totally bounded, meaning that every sequence in the space has a convergent subsequence. In other words, compactness is a stronger condition than completeness.

3. What is the difference between a complete metric space and a closed metric space?

A complete metric space is one in which every Cauchy sequence converges, while a closed metric space is one where the limit of every convergent sequence is also contained within the space. Every complete metric space is closed, but not every closed metric space is complete.

4. What is the significance of function spaces in topology?

Function spaces play a crucial role in topology as they allow for the study of continuous functions between topological spaces. They are also important in the study of metric spaces, where they can be used to define and analyze different types of convergence for functions.

5. Can you give an example of a function space?

One example of a function space is the space of continuous functions between two topological spaces, denoted as C(X,Y). This space consists of all continuous functions from X to Y and can be equipped with various metrics, such as the supremum norm or the uniform norm, to study the convergence of functions.

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