Subspace/induced/relative Topology Definition

In summary, the Subspace/Induced/Relative Topology is a definition used in topology to describe the topology on a subset of a topological space. It is defined by the intersection of the subset with every part of the topology of the original space. This definition may use symbols such as \tau and u, which represent the open sets of the original space, and B, which represents the subset. However, it is important to note that there may be an infinite number of open sets in the original space and the use of u is just for illustrative purposes. Additionally, a U is an element of the \tau, which is standard set-builder syntax for replacement.
  • #1
filter54321
39
0
I'm having trouble understanding the definition of a Subspace/Induced/Relative Topology. The definitions I'm finding either don't define symbols well (at all).

If I understand correctly the definition is:

Given:
-topological space (A,[tex]\tau[/tex])
-[tex]\tau[/tex]={0,A,u1,u2,...un}
-subset B[tex]\subset[/tex]A

The subspace topology on B will be the intersection of B and every part of the topology of A

OR

[tex]\tau[/tex]B={0,B,B[tex]\bigcap[/tex]u1,B[tex]\bigcap[/tex]u2,...B[tex]\bigcap[/tex]un}


...I apologize in advance for my LATEX work.
 
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  • #2
That looks correct idea for the subspace topology... but there is a serious problem in your understanding of topology: you seem to be assuming there are only finitely many open sets.
 
  • #3
The u's are finite for illustrative purposes. I wanted to avoid using the form given by Wikipedia because it has insufficient textual explanation.

They have this definition, but don't specify exactly what a U is:

[tex]\tau[/tex]B={B[tex]\bigcap[/tex]U|U[tex]\in[/tex][tex]\tau[/tex]}

It seems that me that you need to say how U fits into the first topology.
 
  • #4
filter54321 said:
They have this definition, but don't specify exactly what a U is:
Yes they do: they say a U is an element of [itex]\tau[/itex].

This is standard set-builder syntax for replacement: on the right of | you introduce a variable and its domain, and on the left of the | you have a function of that variable indicating what should go into the set you're building.
 
  • #5


I can understand your confusion with the definition of a subspace/induced/relative topology. Let me try to explain it in simpler terms.

A topological space is a mathematical concept that describes the properties of a set and its subsets. It is defined by a collection of open sets, denoted by the symbol τ, that satisfy certain properties (such as being closed under finite intersections and arbitrary unions). In simpler terms, a topological space is a way of defining the "closeness" of points in a set.

Now, a subspace topology is a way of defining a topological space on a subset of a larger topological space. In other words, it is a way of looking at the "closeness" of points in a smaller part of a bigger set. This is done by taking the intersection of the subset with every open set in the larger space. This intersection is denoted by τB, where B represents the subset.

To understand this concept better, let's look at an example. Imagine we have a topological space A, which consists of all the points on a square. Now, we want to define a subspace topology on a smaller subset of this square, let's say a triangle. The subspace topology on this triangle would be the intersection of the triangle with every open set in A. This means that the "closeness" of points in the triangle will be defined by the open sets of the larger space A.

In summary, a subspace/induced/relative topology is a way of defining the "closeness" of points in a subset of a larger set, by using the open sets of the larger set. I hope this explanation helps in your understanding. If you have any further questions, please feel free to ask.
 
  • #6


I can understand your confusion with the definition of Subspace/Induced/Relative Topology. Let me try to provide a clearer explanation for you.

First, let's define some key terms. A topological space is a set A with a collection of subsets, denoted as \tau, that satisfy certain properties. These subsets are called open sets and they represent the "open" or "unrestricted" parts of the space A. The collection \tau is called the topology of A.

Now, a subset B of A inherits a topology from A, which is called the subspace topology on B. This is where the terms "induced" or "relative" come in - the topology on B is induced or derived from the topology on A.

The subspace topology on B is defined as the collection of all subsets of B that can be obtained by intersecting an open set in A with B. In other words, for any open set U in A, the intersection of U with B (denoted as U\cap B) is an open set in B. The collection of all such sets forms the subspace topology on B, denoted as \tau_B.

To summarize, the subspace topology on B is a collection of subsets of B that are open in B, but they are obtained by intersecting open sets in A with B. This allows us to study the topological properties of B within the larger space A.

I hope this explanation helps clarify the definition of Subspace/Induced/Relative Topology for you. Please let me know if you have any further questions.
 

1. What is subspace topology?

Subspace topology is a type of topology that is defined on a subset of a larger topological space. It is created by taking the collection of open sets from the larger space and restricting them to only include points from the subset.

2. How is subspace topology induced?

Subspace topology is induced by the larger topological space. This means that the open sets in the subspace are created by restricting the open sets from the larger space to only include points from the subset. This ensures that the subspace has a consistent and well-defined topological structure.

3. What is the difference between subspace topology and relative topology?

Subspace topology and relative topology are often used interchangeably. However, there is a subtle difference between the two. Subspace topology is induced by a subset of a larger space, while relative topology is induced by an arbitrary subset of any topological space. Essentially, subspace topology is a specific type of relative topology.

4. Can a subspace have a different topology than the larger space?

Yes, a subspace can have a different topology than the larger space. This is because the open sets in the subspace are created by restricting the larger space's open sets to only include points from the subset. This can result in a different collection of open sets and a different topological structure.

5. How is the subspace topology defined mathematically?

The subspace topology is defined mathematically using the concept of open sets. Let (X, T) be a topological space and let A ⊂ X be a subset of X. The subspace topology on A is defined as the collection of all subsets of A that can be written as A ∩ U, where U is an open set in X. In other words, a set U ⊂ A is open in the subspace topology if it can be obtained by taking the intersection of an open set in X with the subset A.

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