Small question to sets

[tired cipher]

You're correct on those two, but try formalizing your reasoning with the epsilon definition that you've been given. This might also help you see how to see if R, Q are open or not

[woven shell]

for (0,1) would the formal be like this?: {x ∈ M| 0 < ε < 1} or would between 0 and 1 a x instead of epsilon 🤔

[tired cipher]

You don't have to try and write out a set definition of (0,1), rather the definition of open asks you to find for any x in (0,1) an eps > 0 such that (x-eps, x+eps) is a subset of (0,1)

So first let x in (0,1) be arbitrary

then eps = min{x,1-x}/2>0 for example (you can in this case find multiple working epsilon) would give you that (x-eps, x+eps) is a subset of (0,1)

here I took epsilon as half the minimal distance that x has with the boundary points 0 and 1

For [0,1] you said it is closed, and the only difference from (0,1) are the points 0 and 1 which are now in the set. So you have to somehow base your argument on what happens or cannot happen if x= 1 or x=0

for Q its denseness should be useful

[woven shell]

.close