Interior Points of Sets: Why is the Interior of Part d.) Empty?

[michonamona]

Hello!

Homework Statement

Find the interior of each set.

a.) {1/n : n\inN}

b.) [0,3]\cup(3,5)

c.) {r\inQ:0<r<\sqrt{2}}

d.) [0,2]\cap[2,4]


I understand that b.)'s interior points are (0,5). I don't understand why the rest have int = empty set.

By definition, if there exist a neighborhood N of x such that N\subseteqS, then x is an interior point of S. So for part d.), any points between 0 and 2 are, if I understand correctly, interior points. But the solution says that part d.)'s set of interior points is an empty set. Why is this?

Thank you

M

Homework Equations

The Attempt at a Solution

 

[ninty]

d) The set is an intersection of two sets. 0 for example is in only one set, but not the other, so it's not in the intersection.

First think about what points are actually in the set, then try to figure out the interior.

 

[lanedance]

it also helps clear on what sets are open.

In this case, I'm guessing the sets are all considered as subsets of R with the usual definition of open sets.

this may seem like a trivial comment, but is important ;)