Intuition for Open Set in Topology

Question

This question is based on:

• Definition of neighborhood and open set in topology
• What is the mathematical distinction between closed and open sets?

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.

Answer 1

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.

Answer 2

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 $T.$