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This question is based on:

Many textbooks and papers base their concepts and definitions on the idea of an open set. I would like to get a solid understanding of its meaning outside of the numbers and its relation to the interval (which makes sense). In thinking about "points" like in a graph, trying to get an intuition for how points can have that same sort of feature as the interval, where $0 < x < 1$ sort of thing, where x can be anything between $0$ and $1$ except $0$ and $1$.

It's hard to imagine a set of points that doesn't have a boundary, because it's a set of points. It seems part of the definition of the set. Otherwise it seems you would have to say the set has 2 types of points, the boundary points and non-boundary points, and so:

  • open set = boundary points + non-boundary points - access rights on boundary points.
  • closed set = boundary points + non-boundary points + access rights on boundary points.

The questions are:

  1. Intuition for an open set in a topology of points that doesn't use numbers or the cyclic definition of open sets as the elements composing a topology.
  2. If discrete topologies (like graphs where vertices are connected by edges) can have open sets, or if it's just a continuous/infinity thing.
  3. Why it's necessary to define a set as open.
Lance Pollard
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    Have you studied metric spaces? – edm May 04 '18 at 03:04
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    "It's hard to imagine a set of points that doesn't have a boundary". It's strange cause you say you understand the situation when there's numbers. Do you understand the sense in which the open interval $(0,1),$ viewed as a subset of the real line, has no boundary? What you write about "access rights", and a boundary being a necessary feature is inscrutable. "Boundary" is a technical term (which is not to say there is no intuition for it). – spaceisdarkgreen May 04 '18 at 03:12
  • I like the idea of an $\epsilon$-neighborhood (first definition). It's like the number line example, where $\epsilon$ is the boundary. So basically the open set is defined in reference to some boundary $\epsilon$, where in the metric space it's the distance (but distance means it's relative to some other node, external to the set of interest). So in a sense it's defined relative to a second set which lies at its boundary, unless I'm misunderstanding. – Lance Pollard May 04 '18 at 03:40
  • closed set: roughly "closed under convergence" – Angina Seng May 04 '18 at 03:43

2 Answers2

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Intuitively, an "open" set is a set with the property that every point of it is "completely surrounded by" other members of the set. This cannot do for a definition since it is too vague. So one usually begins with the idea of a "basic open set" of some sort, from which other open sets can be formed. If there is a metric on the space, then the basic open sets can consist of $\epsilon$-neighborhoods.

Once one has properly defined "basic open set" one can then say that a set $S$ is open means that if $P\in S$ then there is a basic open set $U$ containing $P$ such that $U\subseteq S$.

Then a set is closed if either it has no complement or if its complement is open.

John Wayland Bales
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  • If there isn't a metric space, how do you define a "basic open set" formally or informally – Lance Pollard May 04 '18 at 03:32
  • In specific contexts where one knows exactly what one means by point one tries to decide if nearness makes any sense with regard to pairs of points. If it does make sense but one cannot put a numerical measure on it that satisfies metric properties, then one tries to find sets that, at least, satisfy useful separation properties, such as $T_1$, Hausdorff, etc. – John Wayland Bales May 04 '18 at 03:44
  • Ok cool, thank you! – Lance Pollard May 04 '18 at 03:45
  • A subset $S$ of $X$ cannot have "no complement in $X$". If $S=X$ then its complement $X\setminus S$ is the empty set. And by the definition of a topology the empty set is open. – DanielWainfleet May 04 '18 at 04:04
  • @DanielWainfleet Meaning the same thing in my view. – John Wayland Bales May 04 '18 at 05:53
  • The axioms of set theory imply that the empty set is a thing that exists, whereas "no complement" means that a complement does not exist. We could, I suppose, alter the axioms so that an empty set does not exist. Most of us would regard that as an annoying inconvenience. – DanielWainfleet May 04 '18 at 12:11
  • @DanielWainfleet I'm familiar with the differences of metaphysical opinion in this regard but am more Aristotelean than Platonic in my basic outlook. But these are issues which cannot be reasonably debated in comment fields. Axioms are abstract tools which aid logical consistency. I'll leave it at that. – John Wayland Bales May 04 '18 at 17:19
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Definition. A topology on a set $X$ is a collection $T$ of some, or all, of the subsets of $X$ such that

(i). $\emptyset$ and $X$ belong to $T.$

(ii). If $S$ is a finite subset of $T$ then $\cap S\in T.$

(iii). If $S$ is any subset of $T$ then $\cup S\in T.$

This is a very broad concept. There is at least one topology on any set $X$ because the set $\{\emptyset,X\}$ meets all the requirements for being a topology on $X.$ It is called the coarse topology on $X.$ The set $P(X)$ of all subsets of $X$ is also a topology on $X.$ It is called the discrete topology on $X.$

The members of $T$ are called the open sets of $T.$ The members of $\{X\setminus U:U\in T\}$ are called the closed sets of $T.$ Note that $\emptyset$ and $X$ are each open-and-closed.

With a topology $T$ on $X,$ and $p\in X,$ a neighborhood of $p$ is a set $Y$ such that $p\in S\subset Y\subset X$ for some $S\in T.$ We can show that if $U\subset X$ then $U\in T$ iff $U$ is a neighborhood of each of its members.

Topology was originally called Analysis Situs (Positional Analysis) and was almost entirely concerned with metric spaces, which are certain kinds of topological spaces. The idea was that a neighborhood of a point $p$ should include every point within some distance $r (>0)$ of $p.$

A topological space is a pair $(X,T)$ where $T$ is a topology on $X.$ But it is extremely common to refer to this pair as "the topological space $X$" without specifying which topology on $X$ is actually $T.$

DanielWainfleet
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