Understanding Open and Closed Sets in Topology

In summary, the conversation discusses topology and how it defines open and closed sets. The terms "open" and "closed" refer to subsets of the set X, not the topology \tau. It is possible for a set to be both open and closed, or neither. The example of the set of rational numbers is used to illustrate this concept. The conversation also clarifies that the compliment of a set E is equal to X-E.
  • #1
sol66
60
0
I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears.

So to get to the meat of the question, given that [tex]\tau[/tex] is a topology on the set X giving ([tex]\tau[/tex],X), the members of [tex]\tau[/tex] are called open sets. Up to that point I feel ok, but then the pdf goes to say that the compliment of those members are closed, and so I am guessing that is compliment of that set in reference to X.

So of X is a set of {1,2,3,4,5} and [tex]\tau[/tex] has a particular member that is {1} that the compliment of that would be {2,3,4,5}. And so up to here, hopefully I am understanding the material.

What really gets me is when you start getting sets on your topology which are to be closed, neither open or closed, or even clopen.

Why would you describe a topology to closed rather than open? Isn't being closed suppose to be relative to the members that you have in [tex]\tau[/tex] and the subsets of X which are considered open?

How can a topology be neither open nor closed, I don't get it.

I'm lost, thank you. For your help.
 
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  • #2
Let X be a set, and P(X) the set of all subsets of X. A set [itex]\tau\in P(X)[/itex] is said to define a topology on X if (...I'm sure you know that part already, so I won't type it here). If [itex]\tau[/itex] defines a topology on X we also say that [itex]\tau[/itex] is a topology on X.

The terms "open" and "closed" aren't used about the topology [itex]\tau[/itex]. It's used about subsets of X. A set E is said to be open if [itex]E\in\tau[/itex], and closed if [itex]E^c\in\tau[/itex]. It's possible for a set to be both closed and open. The trivial examples of that are [itex]\emptyset[/itex] and X. It's also possible for a set to be neither, e.g. the set of rational numbers when we take X to be the real numbers and [itex]\tau[/itex] to be the standard topology on [itex]\mathbb R[/itex] (i.e. [itex]\tau[/itex] is the set of all subsets of [itex]\mathbb R[/itex] that can be expressed as a union of open intervals).

You're right that [itex]E^c=X-E[/itex].

I'm not sure if this answers your question. Ask again if it doesn't.
 

Related to Understanding Open and Closed Sets in Topology

1. What is the definition of an open set in topology?

An open set in topology is a set that does not contain any of its boundary points. In other words, for every point in an open set, there exists a small enough neighborhood around that point that is fully contained within the set.

2. How are open sets related to closed sets in topology?

In topology, a set can be both open and closed, or neither open nor closed. Closed sets are the complements of open sets, meaning they contain all the points that are not in the open set. For example, a closed interval [a,b] is the complement of the open interval (a,b).

3. Can an open set be empty?

Yes, an open set can be empty. An empty set is defined as a set with no elements, and since an open set does not contain any of its boundary points, it can be empty.

4. How do you determine if a set is open or closed in a given topology?

To determine if a set is open or closed in a given topology, you need to look at the properties of the topology. A set is open if it satisfies the definition of an open set in that topology. Similarly, a set is closed if it satisfies the definition of a closed set in that topology.

5. Can a set be both open and closed in a topology?

Yes, a set can be both open and closed in a topology. This is known as a clopen set. For example, in the discrete topology, all sets are both open and closed, since every point in the set has a neighborhood that is fully contained within the set, and the complement of the set is also a set in the topology.

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