Are closed sets in a pre-image always contained in an open set in the domain?

In summary, the conversation discusses the existence of an open set in X whose closure is contained in the preimage of an open set in Y. While the definition of a continuous function implies that the preimage of an open set is an open set, it does not guarantee that the closure of the preimage will also be in the preimage. The conversation also considers the role of topology in determining the existence of such an open set, with the example of a metric space being used to illustrate a possible solution.
  • #1
ehrenfest
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1

Homework Statement


This is a topology problem.
I have a continuous map from X to Y, and I take an open set U in Y, and I look at its preimage. Is it true that there must always be an open set in X whose closure is in the preimage of U?

I know that there is always an open set whose closure is in the preimage of the closure of U. But that is not the same thing...

Homework Equations


The Attempt at a Solution

 
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  • #2
The preimage of an open set in Y IS an open set in X. That's basically the definition of continuous function. If it's closure is also the preimage, then the answer is no. Not every open set is also closed. Why are you muddling this up?
 
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  • #3
I am not asking whether the closure of the preimage of an open set is in the preimage, I am asking if there is any open set whose closure is in the preimage.
 
  • #4
Depends on whether your topology guarantees that any open set contains a proper open subset. The preimage is after all, just an open set. It has no special properties beyond that.
 
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  • #5
What if my topology is second countable, locally compact and Hausdorff?
 
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  • #6
If it's a metric space just find an open ball in the set. Call it's radius r, and take the ball of radius r/2 as your set.
 
  • #7
I see. Thanks.
 

FAQ: Are closed sets in a pre-image always contained in an open set in the domain?

What is a closed set in a pre-image?

A closed set in a pre-image is a subset of the pre-image that contains all of its limit points. In other words, if a sequence of points in the pre-image approaches a point in the set, that point must also be in the set.

How is a closed set different from an open set?

A closed set and an open set are two types of subsets in a pre-image. The main difference is that a closed set contains all of its limit points, while an open set does not necessarily contain its limit points.

Can a set be both open and closed in a pre-image?

Yes, a set can be both open and closed in a pre-image. This type of set is called a clopen set. An example of a clopen set is the entire pre-image itself, as it contains all of its limit points and is also open since it does not have a boundary.

How do closed sets relate to continuity in a pre-image?

Closed sets play a crucial role in defining continuity in a pre-image. In a continuous function, the pre-image of a closed set will always be a closed set. This means that the function preserves the property of being closed.

Can a pre-image have an infinite number of closed sets?

Yes, a pre-image can have an infinite number of closed sets. In fact, there are infinitely many closed sets within any pre-image, as any subset of a set can be considered as a closed set if it contains all of its limit points.

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