- #1

sa1988

- 222

- 23

## Homework Statement

Determine whether the following subsets are open in the standard topology:

a) ##(0,1)##

b) ##[0,1)##

c) ##(0,\infty)##

d) ##\{x \in (0,1) : \forall n \in \mathbb{Z}^{+}## ##, x \not= \frac{1}{n}\} ##

## Homework Equations

## The Attempt at a Solution

a) ##(0,1)## is open because for any ##x## in the given interval ## 0 < x < 1 ##, it always has reachable neighbourhoods without any boundary.

b) ##[0,1)## is not open because the point ##x=0## has neighbourhoods outside the set. The set is also not closed because its complement gives ##(-\infty , 0]## and ##(1,\infty )## where the former is not open for similar reasoning.

c) ##(0,\infty)## is open, because all values ##x## in ##0 < x < \infty ## have neighbourhoods, similar to part a).

d) ##\{x \in (0,1) : \forall n \in \mathbb{Z}^{+}## ##, x \not= \frac{1}{n}\} ## is not open because there are discontinuities every time one reaches a point ##\frac{1}{n}##. For this reason the set is also not closed, because the complement will leave 'opposite' discontinuities.As with my previous thread on basic set questions, I'm hoping I've got these right but am running it by the forum because Topology and set theory in general is still currently fairly new territory for me.

Many thanks.