# Baire Category Theorem: Question About Countable Dense Open Sets

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• mr.tea
In summary, the Baire Category Theorem states that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space. This can be proven by considering a countable collection of dense open sets and showing that for any point in the metric space and any positive distance, there exists a closed ball with radius less than 1 that is contained in the intersection of the open sets and also contains the point. This proves that the intersection is dense in the metric space.
mr.tea
Hi,
I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space.
Assume that we have the countable collection of dense open sets ## \{U_n\} ## in a complete metric space ##X##, and let ##x \in X, \epsilon>0##. Since ##U_1## is dense in ##X##, there is ##y_1\in U_1## with ##d(x,y_1)<\epsilon##. Also, as ##U_1## is open, there is ##r_1>0## with ##B(y_1;r_1)\subset U_1##. Then, we can arrange ##r_1<1## such that ##\overline{B(y_1;r_1)} \subset U_1\cap B(x;\epsilon) ##.
Now my question is why we can arrange that the closure will be contained in each of them? I think intuitively it sounds correct, but I didn't succeed to prove it rigorously. Can you please help me here?

Last edited:
there is a typo in your statement, the intersection is dense, not open.

mr.tea
mathwonk said:
there is a typo in your statement, the intersection is dense, not open.
Sorry, fixed it.

i guess you mean by "each of them" the two open sets in your discussion. this is trivial. if a point lies in an open set in a metric space then it has some positive distance from the outside of that set, and hence the ball of radius 1/2 that distance has closure entirely contained in that set.

mr.tea

## 1. What is the Baire Category Theorem?

The Baire Category Theorem is a fundamental result in topology that states that in certain topological spaces, a countable union of nowhere dense sets cannot be the entire space. In other words, in these spaces, the "large" sets must contain at least one "small" open set.

## 2. How is the Baire Category Theorem used in mathematics?

The Baire Category Theorem has many important applications in mathematics, particularly in functional analysis, topology, and measure theory. It is often used to prove the existence of solutions to certain types of equations, as well as to establish properties of functions and spaces.

## 3. Can you provide an example of a space where the Baire Category Theorem holds?

One example of a space where the Baire Category Theorem holds is the real line R with the standard topology. In this space, the Baire Category Theorem can be used to show that a countable union of closed sets with empty interior cannot cover the entire space. This implies that there must be a non-empty open set remaining, which is a key property of the Baire Category Theorem.

## 4. Is the Baire Category Theorem a difficult concept to understand?

The Baire Category Theorem may seem daunting at first, but with a solid understanding of topology and basic set theory, it can be grasped with some effort. It is a fundamental result in mathematics, and its applications are far-reaching, making it an important concept to understand for scientists and mathematicians.

## 5. How does the Baire Category Theorem relate to other theorems in mathematics?

The Baire Category Theorem is closely related to other important theorems in mathematics, such as the Banach-Steinhaus Theorem and the Uniform Boundedness Principle. These theorems are all fundamental results in functional analysis, and they often build upon each other to prove more complex mathematical statements.

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