Open Base and Cover for Topological Space X: Understanding the Relationship

[de_brook]

An open base {B_i} for a topological space X is the class of open sets in X in which any open set in X is the union of sets in {B_i}.

Please consider the following and tell me if i am wrong

observation
An open cover in X is a subclass of some given open base for X. This then should imply that an open cover for X is a basic open cover contained in some given open base.This is because of the definition above and an open cover is a class of open sets whose union contains X

conclusion
Every open cover for a topological space X is a basic opencover. i am saying that an open cover must be contained in some given open base

 

[CompuChip]

I'm not sure if I'm getting you right, but I think you are wrong.

If B is an open base, it means that any open set U can be written as a union of elements from B.
If C is an open cover, it means that C consists of open sets whose union is X.

Since X is an open set, B is definitely also a cover, so open base => open cover.
However, I don't think the converse is true. For example, C' = {X} is an open cover of X (it consists of open sets whose union contains X) but in a general topology, an open set is smaller than X hence cannot be written as a union of elements from C'.

 

[de_brook]

I'm not sure if I'm getting you right, but I think you are wrong.

If B is an open base, it means that any open set U can be written as a union of elements from B.
If C is an open cover, it means that C consists of open sets whose union is X.

Since X is an open set, B is definitely also a cover, so open base => open cover.
However, I don't think the converse is true. For example, C' = {X} is an open cover of X (it consists of open sets whose union contains X) but in a general topology, an open set is smaller than X hence cannot be written as a union of elements from C'.

I think you didn't get my question. Recall the definition of an Basic open cover. It is an open cover of X contained in some given open base for X, thus i am not saying a basic open cover is an open base, but i mean that "Is an open cover a basic open cover?" since X is contained in the union of sets in the open cover

 

[de_brook]

An open base {B_i} for a topological space X is the class of open sets in X in which any open set in X is the union of sets in {B_i}.

Please consider the following and tell me if i am wrong

observation
An open cover in X is a subclass of some given open base for X. This then should imply that an open cover for X is a basic open cover contained in some given open base.This is because of the definition above and an open cover is a class of open sets whose union contains X

conclusion
Every open cover for a topological space X is a basic opencover. i am saying that an open cover must be contained in some given open base

Can anyone in the house help me with this Observation? I am not sure if there is a mistake i have made.

 

[CompuChip]

I was just thinking about your question, I still think my earlier counter-example holds.
If X is a topological space, then {X} is an open cover, but the whole space needn't be (and in general, isn't) an element of the basis.

 

[de_brook]

I was just thinking about your question, I still think my earlier counter-example holds.
If X is a topological space, then {X} is an open cover, but the whole space needn't be (and in general, isn't) an element of the basis.

If {X} is an open cover, is it not possible to be contained in some given open base? If so we can rightly say that {X} is a basic open cover or what do you think?

 

[Hurkyl]

If {X} is an open cover, is it not possible to be contained in some given open base?

Of course. For example, if the basis you were given was the set of all open sets.

(Of course, it is also possible that it is not contained in your given basis)

 

[de_brook]

Now consider the following,
An open base {B_i} for a topological space X is the class of open sets in X in which any open set in X is the union of sets in {B_i}.

Please consider the following and tell me if i am wrong

observation
An open cover in X is a subclass of some given open base for X. This then should imply that an open cover for X is a basic open cover contained in some given open base.This is because of the definition above and an open cover is a class of open sets whose union contains X

conclusion
Every open cover for a topological space X is a basic opencover. i am saying that an open cover must be contained in some given open base
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[yyat]

The way I understand it:

Suppose you have a topological space X and a certain basis for X. An example would be $$\mathbb{R}$$ with the open intervals (a,b) as a basis.

Say you want a cover of X, but which uses only the basis elements. So in the example this would be a cover by open intervals. This is then a "basic open cover". However, the term only makes sense with reference to a fixed basis, otherwise any cover would be "basic" because the family of all open sets forms a basis.

So:
- Every open cover is "basic" with respect to some basis.
- Not every open cover is "basic" with respect to a specific basis.

 

[de_brook]

The way I understand it:

Suppose you have a topological space X and a certain basis for X. An example would be $$\mathbb{R}$$ with the open intervals (a,b) as a basis.

Say you want a cover of X, but which uses only the basis elements. So in the example this would be a cover by open intervals. This is then a "basic open cover". However, the term only makes sense with reference to a fixed basis, otherwise any cover would be "basic" because the family of all open sets forms a basis.

So:
- Every open cover is "basic" with respect to some basis.
- Not every open cover is "basic" with respect to a specific basis.

Thus ordinarily an open cover is basic but w.r.t some given open bases.
Suppose we now consider a Space in which every open cover is considered. For example the compact space. Then a basis in this case will surely have a finite subclass whose union of sets is the whole space.
` `Are the following statement the same;
Every basic open cover has a finite subcover and
Every open cover has a finite subcover

 

[yyat]

Thus ordinarily an open cover is basic but w.r.t some given open bases.
Suppose we now consider a Space in which every open cover is considered. For example the compact space. Then a basis in this case will surely have a finite subclass whose union of sets is the whole space.
` `Are the following statement the same;
Every basic open cover has a finite subcover and
Every open cover has a finite subcover

What do you think? Give a proof or counter-example.

My opinion: Basic open covers are not very important in point-set topology. You should study more central concepts first, for example:
compactness, connectedness, Urysohn's lemma, locally compact spaces, Stone-Weierstrass theorem, metrization, compactification, Baire's theorem, topological groups

 

[de_brook]

What do you think? Give a proof or counter-example.

My opinion: Basic open covers are not very important in point-set topology. You should study more central concepts first, for example:
compactness, connectedness, Urysohn's lemma, locally compact spaces, Stone-Weierstrass theorem, metrization, compactification, Baire's theorem, topological groups

Thanks. I will do that. I am a beginner in the study of topology. I just study something about openbase , open cover and compactness.