# Uncertain about premise for proof.

In summary, it is not implied that g in Corollary 22.3 has the properties given in Theorem 22.2. However, it can be shown that the function p in Corollary 22.3 does have the properties stated in Theorem 22.2. This is because an arbitrary element of X* can be expressed as g^{-1}(z), which ensures that g is constant on sets of this form. This satisfies the premises of Theorem 22.2.
I am reading §22 of Topology by Munkres, in Theorem 22.2 the function g is said to be constant on each set p^(-1)({y}). However the only explicit property given in Corollary 22.3 to the function g is that it is continuous and surjective, but Theorem 22.2 to g in the proof. Is it implied that g in 22.3 also has the properties given in Theorem 22.2?

No, it isn't implicitly assumed that g or p has that property. However, it can be shown that p does have the correct properties of 22.2. Indeed, an arbitrary element of X* has the form $g^{-1}(z)$. We must show that g is constant on sets of this form.
This is true since by definition x is in $g^{-1}(z)$ if g(x)=z. So all elements in $g^{-1}(z)$ are being sent to z. So g is constant on sets of the form $g^{-1}(z)$. So the premises of 22.2 are satisfied.

Thanks! :)

## 1. What is a premise in a proof?

A premise in a proof is a statement or assumption that is taken to be true in order to reach a conclusion. It serves as the starting point for a logical argument and is essential in the process of proving a theorem or proposition.

## 2. Why is it important to be certain about the premise in a proof?

Being certain about the premise in a proof is crucial because the validity of the entire argument depends on it. If the premise is incorrect or uncertain, then the conclusion will also be unreliable. Therefore, it is important to thoroughly examine and verify the premise before proceeding with a proof.

## 3. How can one determine if the premise for a proof is uncertain?

One way to determine if the premise for a proof is uncertain is to analyze the evidence and assumptions that are used to support it. If there is lack of evidence or a flaw in the assumptions, then the premise may be uncertain. It is also important to consider alternative explanations or counterarguments to the premise.

## 4. What steps can be taken to clarify an uncertain premise for a proof?

If the premise for a proof is uncertain, steps can be taken to clarify it. This can include conducting further research, gathering more evidence, or seeking feedback from other experts in the field. It may also be helpful to re-evaluate the assumptions and reasoning behind the premise and make any necessary adjustments.

## 5. Can a proof still be valid if the premise is uncertain?

No, a proof cannot be considered valid if the premise is uncertain. In order for a proof to be accepted, all of its statements and assumptions must be well-supported and proven to be true. If the premise is uncertain, then the entire argument becomes unreliable and the proof cannot be considered valid.

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