Is Munkres Lemma 81.1 Misinterpreted Regarding Normal Subgroups?

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In summary, Munkres Lemma 81.1 is a mathematical theorem used in topology, also known as the "elimination of isolated points" lemma. It states that any continuous function from a closed interval to itself must have at least one fixed point. This lemma has many applications and is often proven using the intermediate value theorem and the concept of connectedness. It can also be extended to higher dimensions.
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[SOLVED] munkres lemma 81.1

Homework Statement


Please stop reading this if you do not have Munkres.

The statement of this lemma implies that pi_1(B,b_0)/H_0 is a group. That does not make sense to me because H_0 is not necessarily normal in pi_1(B,b_0).

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The Attempt at a Solution

 
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Replace the word subgroup with subset and the statement becomes less confusing. Munkres is only talking about \pi_1(B, b_0) / H_0 as a set of cosets.
 

1. What is Munkres Lemma 81.1?

Munkres Lemma 81.1 is a mathematical theorem that is used in the field of topology. It is also known as the "elimination of isolated points" lemma and is often used in the proof of the Brouwer Fixed Point Theorem.

2. What does Munkres Lemma 81.1 state?

Munkres Lemma 81.1 states that any continuous function from a closed interval to itself must have at least one fixed point, or a point that maps to itself. In other words, there is no way to continuously deform a closed interval into itself without at least one point remaining in the same position.

3. Why is Munkres Lemma 81.1 important?

Munkres Lemma 81.1 is important because it is a fundamental result in topology and has many applications in other areas of mathematics. It is also used in the proof of the Brouwer Fixed Point Theorem, which has implications in economics, game theory, and other fields.

4. How is Munkres Lemma 81.1 proven?

Munkres Lemma 81.1 is typically proven using the intermediate value theorem and the concept of connectedness. It can also be proven using other methods, such as the Brouwer Fixed Point Theorem or the Schauder fixed point theorem.

5. Can Munkres Lemma 81.1 be extended to higher dimensions?

Yes, Munkres Lemma 81.1 can be extended to higher dimensions. In fact, it is often stated in terms of n-dimensional spaces, where n is any positive integer. However, the proof becomes more complex in higher dimensions and may require different techniques.

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