# Munkre's Topology Ch 1 sec. 2, ex. #1:

• Constructive Proofs
• benorin
In summary, the conversation discusses an exercise in Munkres' Topology text and finding a proof for it. The proof is essentially the same as a problem found on proofwiki.org, but the person is having trouble understanding the flow and reasoning behind it. The conversation then provides a step-by-step explanation and alternative way of thinking about the problem. It also encourages the person to try and prove a similar problem on their own.
benorin
Homework Helper
Summary:: Subset of Codomain is Superset of Image of Preimage, and similar proof for subset of domain

I was having a hard time doing the intro chapter's exercises in Munkres' Topology text when last I worked on it, and I just wanted to make sure that there's nothing betwixt analysis and topology I'm missing? It's been a while since college for me, so perhaps I'm just a bit forgetful of things disused for two decades. The exercise I got stuck on was

Munkres' Topology Ch 1 sec. 2, ex. #1:

If ##f: A\rightarrow B## and ##A_0\subset A## and ##B_0\subset B##. (a) show that ##A_0\subset f^{-1}(f(A_0))## and that equality holds if ##f## is injective. (b) show that ##f(f^{-1}(B_0))\subset B_0## and that equality holds if ##f## is surjective.

I even found a proof (proofwiki.org) that is essentially the same problem. Couldn't follow the flow of the proof much at all; there didn't seem to be a rhyme nor reason to the starting and ending points of the proof, why one seemly identical definition differing only in notation to the next line was considered and step worth writing out. I'm typically able to follow even protracted ##\epsilon , \delta -##proofs like in real analysis but this simple, basic algebra type proof is giving me trouble I did not expect. Munkres did mention that students would feel comfortable in the beginning of the chapter and find their expertise evaporating near the middle of it.

You prove these things by showing that ##x\in A_0\implies x\in f^{-1}(f(A_0))##.

So, let us fix ##x\in A_0##. You have to show that ##x\in f^{-1}(f(A_0))##. By definition, this means that ##f(x)\in f(A_0)##. But since ##x\in A_0##, trivially ##f(x)\in f(A_0)## so this is really writing out what the definitions mean.

More direct, if ##x\in A_0##, then ##f(x)\in f(A_0)##, hence ##x\in f^{-1}(f(A_0))##.

Another way to think about this, which I use, is just try to understand in words what is happening: ##f^{-1}(f(A_0))## is the set of all points in the domain that get mapped by ##f## to ##f(A_0)##. Clearly ##A_0## is contained in this set.

Were you able to understand this? If yes, can you try to prove ##f(f^{-1}(B_0))\subseteq B_0## on your own now?

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## 1. What is Munkre's Topology?

Munkre's Topology is a branch of mathematics that deals with the study of topological spaces, which are mathematical structures that describe the properties of geometric objects such as points, lines, and surfaces.

## 2. What is Chapter 1, Section 2, Example #1 in Munkre's Topology?

Chapter 1, Section 2, Example #1 in Munkre's Topology is a specific example or problem that is presented in the second section of the first chapter of Munkre's Topology textbook. It is used to illustrate a concept or theorem in the field of topology.

## 3. What is the purpose of Example #1 in Munkre's Topology?

The purpose of Example #1 in Munkre's Topology is to demonstrate how to apply the concepts and theorems discussed in the textbook to solve a specific problem or scenario. It helps to solidify the understanding of the material and its practical applications.

## 4. Is Example #1 in Munkre's Topology a difficult problem?

This can vary depending on the individual's level of understanding and familiarity with topology. Some may find it challenging, while others may find it relatively easy. However, the purpose of the example is to help the reader grasp the concepts and not to be overly complex.

## 5. Can Example #1 in Munkre's Topology be applied to real-world situations?

Yes, the concepts and theorems presented in Munkre's Topology can be applied to various real-world situations, such as in physics, engineering, and computer science. Example #1 serves as an illustration of this application and helps to bridge the gap between theory and practicality.

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