I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the closure minus the interior, then thats the "rim" of the set or quite literally the boundary. But then when I think of closure in that way it seems like it is saying that since we have to subtract away the interior, then that must mean the closure is the whole entire set including the boundary. If that's the case (which I'm pretty sure it's not) then why do we have to have another name for a set? Again, a visualization would be great if it can be described.
[Math] Difference between closure and the boundary
definitiongeneral-topologyreal-analysis
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Best Answer
Consider the set $[0,1]$ in $\Bbb R$. You can easily check that it’s a closed set: if $x<0$, $(x-1,0)$ is an open nbhd of $x$ disjoint from $[0,1]$, and if $x>1$, $(1,x+1)$ is an open nbhd of $x$ disjoint from $[0,1]$. Thus, $[0,1]$ is its own closure. However, its interior is $(0,1)$, so its boundary is the two-point set $\{0,1\}$, which is certainly not at all the same as $[0,1]$.