Discrete topology, product topology

In summary, we are looking at different subsets of the infinite product space 2^\omega = 2^\mathbb{N} \cup \{ 0 \}. We are interested in determining whether these subsets are open, closed, both, or neither in the product topology. To do so, we need to consider the individual factors which are discrete and check that all but finitely many of them are the whole space. This will tell us whether the subset is open or not. For the difficult task of checking if a subset is closed, we need to look at its complement and determine if it is open or not.
  • #1
mathsss2
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For each [tex]n \in \omega[/tex], let [tex]X_n[/tex] be the set [tex]\{0, 1\}[/tex], and let [tex]\tau_n[/tex] be the discrete topology on [tex]X_n[/tex]. For each of the following subsets of [tex]\prod_{n \in \omega} X_n[/tex], say whether it is open or closed (or neither or both) in the product topology.

(a) [tex]\{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}[/tex]
(b) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n) = 0 \}[/tex]
(c) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}[/tex]
(d) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}| = 5 \}[/tex]
(e)[tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}|\leq5 \}[/tex]
 
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  • #2
Recall that [tex]\omega = \mathbb{N} \cup \{0\}[/tex]
 
  • #3
And what are your thoughts on the problem?
 
  • #4
Here is what I know:

So remember that open sets in the infinite product topology is really just having all but finitely many the whole space and the rest are open. Since the individual factors are discrete, you only need to check that all but finitely many are the whole space.

e.g. in (a) the 10th coordinate has a specific value, but all other coordinates can be whatever, so this is certain open.[tex]f[/tex] is a function. Here [tex]\omega = \mathbb{N} \cup \{ 0 \}[/tex] (the reason for using [tex]\omega[/tex] is because he is using it to refer to the natural as an ordinal, but whatever that is not important). If it helps you can think of [tex]\prod_{n\in \omega} X_n[/tex] as [tex]\prod_{n=0}^{\infty} X_n[/tex]. We define [tex]\prod_{n=0}^{\infty}X_n[/tex] to be the set of all functions [tex]f: \mathbb{N} \to \{ 0 , 1\}[/tex] that satisfies [tex]f(n) \in \{ 0 , 1\}.[/tex]

This is as far as I've gotten.
 
  • #5
There's a nice graphical representation of the product topology on Y^X (i.e. the product of the space Y |X| times). Namely, if we draw X as an "x-axis" and Y as a "y-axis", then elements in X^Y are "graphs of functions" in the X-Y "plane". An open nbhd of an element f is the set of all functions g whose graphs are close to the graph of f at finitely points. We get different nbhds by varying the closeness to f and/or the set of finite points.

In our case the product space is 2^w=2^N, whose "plane" looks like two copies of the naturals N. In other words, if you were to imagine this as a 'subset' of R^2, it's just the set [itex]\{(n,i) \colon n \in \bN, i \in \{0,1\}\}[/itex].

Maybe this will help you.
 
  • #6
Progress:

Take set (b). Let [tex]B = \{f \in \prod_{n \in \omega} X_n | \;\exists n \in \omega \; f(n) = 0 \}[/tex]. If [tex]f\in B[/tex] then there exists m such that f(m)=0. Then the set [tex]\{g \in \prod_{n \in \omega} X_n |\; g(m) = 0\}[/tex] is an open neighbourhood of f contained in B. Therefore B is open.

It's usually more difficult to check when a set is closed. You have to look at its complement and decide whether that is open. Sometimes this is straightforward. For example, the complement of set (a) is the set of all f such that f(10)=1. That is open, so set (a) is closed as well as open.

For a slightly less easy example, look at set (c). Let [tex]C = \{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}[/tex]. If [tex]f\notin C[/tex] then there exists m such that f(m)=f(m+1)=0. Then [tex]\{g \in \prod_{n \in \omega} X_n |\; g(m) = g(m + 1) = 0\}[/tex] is an open neighbourhood of f containing no points of C. Therefore the complement of C is open and so C is closed.

I still do not know how to do parts (d.) and (e.)
 

1. What is discrete topology?

Discrete topology is a type of topology in mathematics that defines a set of points where every point is an open set. This means that every subset of the set is considered an open set, making it a very specific and simple type of topology.

2. How is discrete topology different from other types of topology?

Unlike other types of topology, such as the Euclidean topology, discrete topology does not rely on distance or proximity between points. Instead, discrete topology focuses on the concept of open sets, where every point is considered an open set and every subset is also an open set.

3. What is product topology?

Product topology is a type of topology that is formed by combining two or more topological spaces. It is created by taking the Cartesian product of the individual spaces and defining a topology on the resulting product set based on the open sets of the original spaces.

4. How is product topology related to discrete topology?

Product topology can be used to construct discrete topology by taking the Cartesian product of a set with itself. This means that the resulting topology will have open sets that contain only one point, making it a discrete topology. In this sense, product topology can be seen as a way to generalize and create different types of topologies, including discrete topology.

5. What are the applications of discrete topology and product topology?

Discrete topology and product topology have various applications in mathematics, computer science, and other fields. They are commonly used in the study of topological spaces, algebraic topology, and functional analysis. In computer science, product topology is used in the design of computer networks and distributed systems. Discrete topology is also used in graph theory for analyzing discrete structures and networks.

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