# Topology: Munkres - Urysohn lemma

• Fisicks
In summary, we are trying to show that if A is a closed G(delta) set in a normal space X, then there exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A. In order to prove this, we define f_n using the Urysohn lemma and show that f is continuous using the fact that each f_n is continuous.
Fisicks
Hi, the problem I am referencing is section 33 problem 4.

Let X be normal. There exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A, if and only if A is a closed G(delta) set in X.

My question is about the <= direction.

So let B be the collection of open sets whose intersection is A and index them with the natural numbers. Let U_n be an element of B. For each U_n, define a function f_n. To define f_n follow the proof of Urysohn lemma using A=A, B=X-U_n.

Define fx= sup{f_n(x)} for all n.

Clearly fx=0 iff x is in A. My problem is with showing continuity. Part of me thinks that if x is an element of X and (a,b) is a basic open set of fx, then there exists an open set U such fU is contained in (a,b) since each f_n is continuous.

I think we need a reference here as Urysohn doesn't make a statement about the values of ##f## outside ##A## and ##B##. Also "G(delta)" needs to be explained.

However, I am not sure how to show this.

Hi there,

Thank you for your post. I understand your question and will try to provide an explanation.

First, let's define the function f_n as follows:
f_n(x) = d(x, X-U_n) / [d(x, X-U_n) + d(x, A)]

where d(x, X-U_n) represents the distance from x to the complement of U_n in X, and d(x, A) represents the distance from x to the set A.

Now, let's take an arbitrary point x in X and an open interval (a,b) containing fx. We need to show that there exists an open set U containing x such that f(U) is contained in (a,b).

Since f_n is continuous, for each n, there exists an open set V_n containing x such that f_n(V_n) is contained in (a,b). Now, let's define U as the intersection of all V_n's. Since U is an intersection of open sets, it is also open.

For any point y in U, we have d(y, X-U_n) < d(y, X-U_n) + d(y, A) for all n, which means that f_n(y) > 0 for all n. Therefore, sup{f_n(y)} > 0.

Now, let's consider the point x itself. Since x is in U, it must also be in V_n for all n. Therefore, f_n(x) is contained in (a,b) for all n. This means that sup{f_n(x)} is also contained in (a,b).

Hence, we have shown that for any x in X and any open interval (a,b) containing fx, there exists an open set U containing x such that f(U) is contained in (a,b). This proves the continuity of f.

I hope this helps. Let me know if you have any further questions.

## 1. What is the Munkres-Urysohn lemma?

The Munkres-Urysohn lemma is a theorem in topology that states that for any two disjoint closed subsets in a topological space, there exists a continuous function that separates them. In other words, the function takes on different values for points in each of the two subsets.

## 2. What is the significance of the Munkres-Urysohn lemma?

The Munkres-Urysohn lemma is an important tool in proving separation properties in topology. It is commonly used to prove the normality of topological spaces, which is a key concept in various areas of mathematics, such as analysis and geometry.

## 3. How is the Munkres-Urysohn lemma proved?

The Munkres-Urysohn lemma can be proved using the Tietze extension theorem, which states that any continuous function defined on a closed subset of a normal topological space can be extended to the whole space while preserving continuity.

## 4. Can the Munkres-Urysohn lemma be generalized?

Yes, the Munkres-Urysohn lemma can be generalized to the Urysohn's lemma, which states that for any two disjoint closed sets in a normal topological space, there exists a continuous function that takes on values in a specified closed interval, separating the two sets.

## 5. How is the Munkres-Urysohn lemma used in practical applications?

The Munkres-Urysohn lemma has various applications in mathematics and physics, such as in the study of electromagnetic fields and the construction of fractals. It is also used in computer science and engineering to model and analyze complex systems.

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