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Let $\mathbb{R}_{[-)}:=\{[a,b);a,b\in \mathbb{R}\}$ a topology and $(0,1)$ be an intervall. I want to determine the interior and boundary of the $(0,1)$ in this topology.

The interior is $(0,1)$, but I have problems with the boundary.

Lets consider the point $1$. Would the element $[1,2)$ of $\mathbb{R}_{[-)}$ be a neighbourhood of the element $1$? I don't think so, because I can't inscribe an open set into $[1,2)$ which contains the point. (My solution says, this is a neighbourhood of $1$ which is why I am confused now).

I would think that the point $1$ is a boundary point, but this must be wrong according to the solution.

3 Answers 3

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The closure of $(0,1)$ is $[0,1)$: every neighbourhood of $0$ contains one of the form $[0,x)$ which will always intersect $(0,1)$, if $x < 0$, then $[x,0)$ is a neighbourhood of $x$ that misses $(0,1)$, and also if $x \ge 1$, $[x,x+1)$ is a neighbourhood of $x$ that misses $(0,1)$ as well. So we only add $0$ to the already open set $(0,1)$. So the boundary is only $\{0\}$ ,the closure minus the interior.

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Every interval $[1,a)$ (for $a>1$) is open in this topology, so yes, $[1,2)$ is a neighbourhood of $1$ and no, $1$ is not in the boundary of $(0,1)$.

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    Then I don't understand why it is open. How can I find an open ball around 1, for which $U_\epsilon (1)\subset [1,a)$? – 2017-02-07
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In your topology, $[1,2)$ is a neighborhood of $1$, but $1$ isn't in the set $(0,1)$. The neighborhood of $1$ doesn't intersect the set, so $1$ isn't a boundary point. What point(s) have the property that every open set containing it/them intersects $(0,1)$?

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    I am somehow already stuck in that I don't understand why $[1,2)$ is open in this topology. If i look for an open ball around 1, which elements are contained in this open ball then? Can it only be elements of the form $[a,b)$, or single points too? I don't understand what counts as subset of this topology. Can it be single points too? – 2017-02-07
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    @EpsilonDelta: The set $[1,2)$ is open by definition (the sets of the form $[a,b)$ are open). – 2017-02-08