Understanding Clopen Sets in X: A Wikipedia Example

In summary, the conversation discusses the concept of a clopen set in a topological space. The example of the union of two intervals is used to show that in such cases, the components will be clopen. The definition of boundary is also discussed, with the conclusion that in the subspace topology, the set [0,1] is clopen because its boundary is empty in X, even though it is not in R.
  • #1
theneedtoknow
176
0
This isn't really a homework question, can someone just explain this bit from wikipedia?

consider the space X which consists of the union of the two intervals [0,1] and [2,3]. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set [0,1] is clopen, as is the set [2,3]. This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

and later:

A set is clopen if and only if its boundary is empty.

Ok...so take the set [0,1] C X where X = [o,1]U[2,3]...how is the boundary of [0,1] empty? Isn't the boundary of [0,1] the 2 points 0 and 1? So I don't really get how [0,1] is clopen in this case
 
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  • #2
What is the definition of boundary? Remember this is in the subspace topology and you shouldn't just think that your intuition about [0,1] being a subset of R is correct - after all [0,1] is open and closed...
 
  • #3
Is 0 really in the boundary of [0,1]? By definition, it is so if every open set U of X containing 0 contains points of [0,1] and of X\[0,1]=[2,3]. Well, take for instance the open set (-1,1)nX=[0,1). It does not contain points of [2,3], so 0 is not in the boundary of X.

What happens here is that [0,1] has boundary {0,1} in R, but not in X.
 
  • #4
Ah ok, thanks guys :) its more clear now
 

1. What is a clopen set?

A clopen set is a type of subset in a topological space that is both open and closed. This means that the set includes all of its boundary points and does not have any points on its boundary that are not included. In other words, the set is both open and closed at the same time.

2. How do clopen sets relate to topology?

Clopen sets are an important concept in topology, which is the study of spaces and their properties under continuous transformations. In particular, clopen sets are used to define and characterize different types of topological spaces, such as connected and compact spaces.

3. Can you give an example of a clopen set?

One example of a clopen set is the set of all integers in the real line. This set is both open and closed since it contains all of its boundary points (the integers) and does not have any points on its boundary that are not included.

4. What is the significance of clopen sets in X?

In the context of the Wikipedia example, X refers to a specific topological space. Understanding clopen sets in X can help us better understand the properties and behavior of this space, and can also be applied to other topological spaces.

5. How are clopen sets used in real-world applications?

Clopen sets have various applications in mathematics and science, including in the fields of topology, analysis, and differential equations. They can also be used in real-world applications such as data analysis and machine learning, where they are used to define and characterize different types of data sets and their properties.

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