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Introduction of Baire Category Theorem

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  • Last Updated : 21 Jun, 2021
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Introduction of Baire Category Theorem :
Baire’s category theorem, often known as Baire’s theorem and the category theorem, is a conclusion in analysis and set theory that says that the intersection of any countable collection of “big” sets stays “large” in certain spaces. The use of the word “category” in the name alludes to the theorem’s interaction with the ideas of first and second category sets.

To put it another way, if a space S is either a full metric space or a locally compact T2-space, then the intersection of any countable collection of dense open subsets of S must be dense in S.

Assume that no Fk has a nonempty open set. Then, and only then, no Fk equals E.
Because F1 6= E, F1 is a nonempty open set that must include one element. The open is not included in the set F2. B(x1;1/2) ball. As a result, the nonempty open set F2 B(x1;1/2) contains an open ball.

Using the inductive definition principle, we obtain a series of open balls Bk = B(xk;k) such that, for all integers (k 1, 0 k)

Bk+1 = B(xk;k/2), and Bk Fk = The family (Fk)kN, in particular, must be infinite. (In other words, the proof is complete in the finite case.) Because, for n m,

Because there are complete metric spaces that are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces that are not metrizable, neither of these statements implies the other (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in functional analysis; the uncountable Fort space).

The concept of countability, as a way of comparing sets with the set of natural numbers, is frequently taught early in undergraduate real analysis courses. We know that countable sets include the set of integers, the set of odd integers, and the set of rationals.

Uncountable sets are defined as sets that are not countable, such as the set of all irrationals. A set is countable or uncountable depending on whether it has a one-to-one relationship with the natural numbers.

By definition, a metric space is a set with a distance function. Because there are no other restrictions on the set, the concept of category may be extended to a wide range of metric spaces, including Euclidean spaces, function spaces, and sequence spaces.

Stanislaw Mazur, a Polish mathematician, proposed the following game in 1935 :

Player 1 and Player 2 are the names of the two players. A subset A of the interval [0, 1] is determined ahead of time, and the participants pick subintervals alternately. In [0, 1] such that In+1 In for each n is greater than 1. Player 1 wins if the intersection of all In intersects A, and Player 2 wins if the intersection of all In intersects A.

This intersection may be forced to be disjoint from A
There are several ways to state the Baire category theorem. We provide five variants of this theorem and their equivalence.

  1. Each interval [a,b] represents a set of the second category.
  2. R belongs to the second group.
  3. R’s residual subsets are all dense.
  4. There is an empty interior in any countable union of closed sets with an empty interior.
  5. A dense intersection is any countable intersection of open dense sets.

Remark : The Baire category theorem is a “pretty deep finding,” — as you can see, it isn’t (the proof of the equivalence of the three concepts of compactness was more difficult).

But what is profound is the simple notion of considering countable unions. Thick setups from nowhere This was a brilliant idea on Baire’s (and Osgood’s) part, and it worked.

Applications :
1. Show that for every k, y lies in BX(xk, rk/2). (Hint: For p = 0, y is the limit of (xk+p).
Solution :
As seen above, y lies in BX(xk, rk) and hence in Uk for every k.
In other words, y is contained within G. We also see that y is in BX(x, r/2) since each xk belongs to this closed set. As a result, y also exists in BX(x, r). This demonstrates what we want to demonstrate.
This result is frequently used in applications in the following format. Let Xn be a sequence of closed sets in a full metric space (X, d) such that X = nXn; that is, X is the union of the sets Xn. We then assert that at least one Xn’s interior is not empty. This is demonstrated by the following paradox.
Assume Xn has an empty interior for every n. As a result, the complement Un = X Xn$ of Xn is open.

2. The set is dense. In the reals, the set of all rationals Q is dense: In R, let a b be. Then there’s a logical number in there somewhere (a, b).
Let ∈  = b -a.
When 1 N b a, choose N such that N > 1 ba.
Assume A = m N : m N is a subset of Q. We assert that A (a, b) 6=. Assume the opposite is true. Then we can choose m1, which is the biggest integer, such that m1 N a. If m+1| N > b, then m+1| N > b. 
But then b a m1 + 1 N m1 N = 1 N b a, which is a contradiction. As a result, (a, b) Q 6=.

Assume Apseparates points of R in Lemma 1 :
The graph of pis was then closed in R2. Proof. If this is not the case, then there is a series of points in R tending to x0as n such that(3.1) p(xn)y0as n and y06=p (x0). For each f Ap,f and f(p) So, for every f Ap, (3.2) f(p(x0)) = limn f(p(xn)) = limn f(y0). This indicates that Ap does not separate rand points, completing the proof of the lemma.

The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are well recognised to be related. We investigate the exact relationship between these two theorems. Using a broad concept of logic, we demonstrate that the traditional Omitting Types Theorem applies for a logic if a certain related topological space has all closed subspaces Baire. We also look at greater Baire category requirements, and therefore stronger Omitting Types Theorems, as well as a game variant. We build abstract logic using instances of spaces already explored in set-theoretic topology to show that the game Omitting Types assertion is consistently not equal to the classical one.

Conclusion :
Given a linear space E and a countable family (Pk) of seminorms on E that satisfy (b) and (c), we can topologize E as a Frechet space in only one way.
Thus, this was the end for the mere introduction of Baire Category Theorem. Hope this article helped you in getting a gist for this topic and makes you dwell in detail!

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