Is there a special name given if a number is on the edge of an interval?

Question

For instance, if I have the interval $[0,1]$, is it proper to say that $0$ is on the lower bound of the interval?

I'm looking for the most elegant way to express this.

Answer 1

One relevant term here is "boundary point." Roughly speaking, $x$ is a boundary point of $Y$ (I'm working in $\mathbb{R}$ for simplicity) iff there are points outside of $Y$ "arbitrarily close" to $x$: for all $\epsilon>0$, there is some $z$ such that $z\not\in Y$ but $\vert x-z\vert<\epsilon$. For example, $x=0$ is a boundary point of $Y=[0,1]$ since given $\epsilon>0$ the point $-{\epsilon\over 2}$ is within $\epsilon$ of $x$ but is not in $Y$.

(There is a more general notion of "boundary point," but it's more technical.)

However, this is a somewhat slippery notion. For example, every real number is a boundary point of $\mathbb{Q}$, since there are irrationals arbitrarily close to every real number. Similarly, $0$ is also a boundary point of $(0,1)$ even though $0\not\in (0,1)$. For this reason, terms like "endpoint" may be preferred.

(Why not "minimum" or "maximum?" Well, those terms are great for individual intervals, but consider something a bit more complicated like $A=[0,1]\cup [2,3]$. The point $1$ is an endpoint of $A$, but is neither the minimum nor maximum of $A$.)

Answer 2

If the word is about a single/individual interval (on the real line),
I would just use the terms lower end and upper end to keep things simple.

I am pretty sure I've seen these being used.

So for $[0,1]$ the number zero is the lower end,
and the number one is the upper end of the interval.

Answer 3

Let $(x,y)$ denote the set $\{\alpha\in\Bbb{R}\mid x<\alpha<y\}$ (open interval)

Let $(x,y]$ denote the set $\{\alpha\in\Bbb{R}\mid x<\alpha\le y\}$ (half open)

Let $[x,y)$ denote the set $\{\alpha\in\Bbb{R}\mid x\le\alpha< y\}$ (half open)

Let $[x,y]$ denote the set $\{\alpha\in\Bbb{R}\mid x\le \alpha\le y\}$ (closed)

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There are a few different terms used to refer to different extreme points along subsets of the real line:

A lower bound of an interval $I$ is any element $u$ such that for any $e\in\Bbb{R}$, if $e<u$ then $e\notin I$. For example, $-14$ is a lower bound of $(0,1)$. Note that a lower bound does not need to be close to the "lower end" of the interval.

An upper bound of an interval $I$ is any element $u$ such that for any $e\in\Bbb{R}$, if $e>u$ then $e\notin I$. For example, $96$ is an upper bound of $(0,1)$. Note that an upper bound does not need to be close to the "upper end" of the interval.

The minimum of an interval is its least element - i.e. $x$ is the minimum of $[x,y)$ and $[x,y]$. Note that sets which are open on the left do not have a minimum.

The infimum of an interval (open or closed) is its greatest lower bound - i.e. the infimum of all of the above intervals is $x$. Note that every minimum is an infimum, but the converse need not hold.

The maximum of an interval its greatest element - i.e. $y$ is the maximum of $(x,y]$ and $[x,y]$. Not that intervals which are open on the right do not have a maximum.

The supremum of an interval (open or closed) is its least upper bound - i.e. the supremum of all of the above intervals is $y$. Note that every maximum is an supremum, but the converse need not hold.

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In the Euclidean topology on $\Bbb{R}$, every limit point of an interval is either an infimum or supremum. It should be noted, however, that limit points in general (even when working in the reals) do not need to behave this way. For a full discussion of limit points, I would suggest looking through any suitably thorough undergraduate textbook on topology.