Open Sets in Topological Spaces: Understanding U=intcl U

In summary, the conversation discusses regular open sets and the question of whether U=〖int〗_X 〖cl〗_X U holds for all open sets in a topological space. The speaker asks for help finding a counterexample and also expresses confusion about choosing subsets for a closure function satisfying Kuratowski closure axioms.
  • #1
99butterfly
4
0
Regular open sets,,,,

If U is an open set in a topological space (X,τ),is it true that U=〖int〗_X 〖cl〗_X U?Justify.
 
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  • #2


please help me with this question...

I think this says about regular open sets.
so I need to find an open set which does not satisfy the equality given in the question above.
 
  • #3


Try to find a counterexample. Take an nice open set in a nice space and remove a point.
 
  • #4


thank you verymuch micromass...

I have another question regarding closure axioms.

I know all the axioms but I'm confused with choosing two arbitrary subsets of X since it takes two possibilities for theta.

Please somebody help me with this!

Let θ:P(X)→P(X),where θ(A)={A ;if |A| <|N|
X ; O/W.
Verify that θ satisfy Kuratowski closure axioms.
 
  • #5


Well, what are the axioms?? Which ones are troubling you??
 

1. What is the meaning of "U=intcl U: Justify Answer"?

"U=intcl U: Justify Answer" is an equation often used in computational chemistry and physics. It represents the electronic energy of a system, where U is the potential energy and intcl is the energy from the interaction between particles. The phrase "justify answer" means that the value of U=intcl U is being used to explain or support a scientific argument or conclusion.

2. How is the equation U=intcl U used in scientific research?

U=intcl U is often used to calculate the total energy of a chemical or physical system. This information can be used to predict how the system will behave and to make theoretical and experimental predictions about its properties.

3. What does the subscript "intcl" in U=intcl U represent?

"intcl" stands for "interaction of particles" and it is a measure of the attractive and repulsive forces between particles in a system. In the context of U=intcl U, it represents the energy contributed by the interactions between particles in a system.

4. Why is it important to justify the answer when using the equation U=intcl U?

Justifying the answer when using U=intcl U is important because it provides a sound scientific basis for any conclusions or predictions made using the equation. By using this equation, scientists can ensure that their calculations are accurate and supported by scientific evidence, making their research more reliable and credible.

5. Are there any limitations to using the equation U=intcl U in scientific research?

Like any scientific equation, U=intcl U has its limitations. It is based on certain assumptions and simplifications, and may not accurately represent all systems or situations. Additionally, the accuracy of the calculations may be affected by the quality of the data and the complexity of the system being studied. Scientists must carefully consider these limitations when using U=intcl U in their research.

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