user777

Hello, I have a question in topology, I have this definition that says the following: If B is a basis for a topology on X, the topology T generated by B is described as follows: U \in T is said to be open if for all x \in U, there exists B* \in B s. t. x \in B* and B \subseteq U.

I took an example: Let X = {a, b, c} and let B be the basis which is: B = { {a}, {b}, {c}}. Now, I just applied the definition to get a topology T, so I got T = { {a}, {b}, {c}} but T is not a topology here since neither X nor phi are subsets of T. Do I have something wrong

Hi! I have a question: First from this article:

https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/

This new article states that the paper by David Asperó and Ralf Schindler show Martin’s Maximum and Woodin’s axiom implies each other. Note that if one of them is true, then continuum hypothesis is false.

My question: Why don't we add Woodin's axiom or Martin’s Maximum in ZFC axioms and show that continuum hypothesis is false and we are done. Why don't we say that continuum hypothesis is no longer true? What is left to show?

Hi, I read an article that said in the first paragraph the following:

Randomness is powerful. Think about a presidential poll: A random sample of just 400 people in the United States can accurately estimate Clinton’s and Trump’s support to within 5 percent (with 95 percent certainty), despite the U.S. population exceeding 300 million. That’s just one of many uses.

My question is: how did he know that from sample of 400 persons, then we can know with 95% certainty that 5% of the U.S. populations are supporting the ones who have the majority in the sample?

Hi guys! I have a question: Juris Hartmanis in his paper: "On Computational Complexity and the Nature of Computer Science" states, "A considerable part of our early work on complexity theory was dedicated to showing that we had defined a meaningful classification of problems according to their computational difficulty and deriving results about it. We showed that our classification was robust and was not essentially altered by minor changed in the model and that the complexity classification indeed captured the intuitive ideas about the complexity of numbers and functions." and said later, …