what exactly is an open set?

Question

Many, infact all the books on topology I have come across define open sets in the following way:

"A set $A$ is said to be open if by moving in small amounts in any direction about any point we land up at a point which belongs to the same set."

Is it so that an open set is always a collection of points only? OR does there exist a general definition of open sets, without taking points into consideration?

Answer 1

The "small amounts in any direction" idea doesn't have any direct translation to topology, but another similar idea has an exact definition in topology: here, open sets are intuitively those sets which surround all the points they contain.

The justification of this is as follows:

Start with any topological space and two subsets $A$ and $B$ inside that space. Now in a plain old set, either $A$ and $B$ intersect or they do not. However, in a topological space, we can formalize the idea that $A$ and $B$ 'touch', if not actually intersect. Say that $A$ and $B$ 'touch' if every open set containing $A$ intersects $B$ or every open set containing $B$ intersects $A$ [for future reference: this happens iff 'the closure' of the two sets intersect in the usual sense].

For example, on the real line, $A = [0,1)$ 'touches' $B = [1,2]$. Why? Because any open interval containing $B$ will spill over enough to detect an intersection with the nearby set $A$.

Back to the idea of open sets as surrounding sets. By definition, any point inside an open set $U$ automatically does not 'touch' anything outside that set because by definition the open set $U$ is proof that it doesn't!

This gives a (admittedly rather vague) sense that a point in an open set is spatially separated from the points outside that open set.

Answer 2

"moving in small amount" is relevant only to metric spaces.

For a topological space $X$ with topology $\tau$, $A$ is open if $A \in \tau$. Meaning, you define the topology by defining what is an open set in the topology.

For example, if you define a topology $\tau = \{\phi, \{a\}, \{a,b\}\}$ over a finite space $ X = \{a,b\}$, $A$ is open iff $a \in A$ or $A$ is empty.

Answer 3

If you have a "space" or "ground set" $X$ with points $x$ then any subset $A\subset X$ is a collection of points $x\in X$ $-$ there is no way around that. If some such sets $A$ are declared "open" they necessarily "consist" of points $x$, and if one wants to describe what "openness" for a set $A$ means one is forced to talk about points $x\in A$ and $\notin A$.

In order to get a feeling for "openness" one has to talk about "neighborhoods". It is the very essence of the notion of "topology" on a set $X$ that each point $x\in X$ possesses a system ${\cal V}(x)$ of neighborhoods $V$ of $x$. A neighborhood of $x$ is a preferentially small set $V\subset X$ that contains $x$ and ${\it all}$ points $x'\in X$ which are "sufficiently near" $x$. What "sufficiently near" means is described by axioms (which are obviously fulfilled if nearness is defined by a metric), e.g., if $V$ is a neighborhood of $x$ then $V$ is also a neighborhood of all points $x'$ "near" $x$.

It is essential that ${\cal V}(x)$ contains "arbitrarily small" sets, like balls of radius ${1\over n}$ for arbitrarily large $n$.

Having all this in mind, an arbitrary subset $A\subset X$ is declared open, if $A$ is a neighborhood of each of its points $x$. This means that for each point $x\in A$ the set $A$ contains all points $x'\in X$ that are sufficiently near $x$.

Answer 4

One could characterize open sets as sets whose points cannot be approached from outside the set. That's probably more of a motivation for a definition than a definition in its own right.