Intuition behind open set in topology

Question

I am reading Munkres Topology Chapter 13, in which some examples of bases of topologies are given. One of the examples compares the two possible bases (a, b) and [a, b) on the real line.

I understand that both (a,b) and [a,b) satisfy the definition of basis (intersection between two element has the same form and all elements together form the whole real line).

However, in the definition of a topology (as a subset of the power set of the real line), every element is called an open set. Clearly this "open" definition is different from the traditional "open" definition as [a, b) cannot be open in the traditional definition.

Furthermore, I can see that even [a, b] can be "open" in the real line (in the topology generated by [a, b]) because any non empty intersection also has the form [a, b] and the whole union makes up the whole real line. But this makes no sense using the traditional definition where [a, b] is obviously closed.

My question is why every element of a topology is named "open"? What is the intuition behind this name?

Answer 1

I believe that this terminology predates topological formalism. When nineteenth century mathematicians began laying the foundations of what we call today analysis and topology, they noticed that what we call today open sets in the standard topology on $\mathbb R$ have nice properties (closed under unions, finite intersections, etc...).

When considering other spaces besides $\mathbb R$, it is useful to determine the least amount of structure to apply to that space to specify its unique properties. Consider, for instance the circle. It has properties that are different from any subset of the real numbers. How much information do we need to know about a space before we can declare that it has the properties of a circle? Certainly this depends on which properties in which you are interested, but early topologist discovered that not only can many interesting properties can be determined by simply specifying which subsets of that space have the same behavior as open sets in $\mathbb R$, but also that proving that these properties hold is not terribly difficult in most cases. So by specifying a relatively small amount of information about the space, we can describe it in great detail with relative ease.

I believe that the name "open set" simply carries over from the standard topology on the reals where the description of open relates more clearly with the English definition of the word. Nevertheless, as I often have to explain to my non-mathematician friends, if an English word has a mathematical definition associated with it, that definition need not have any relation to the English definition of the word (though it usually does to some extent). So even though we call sets open that can't be described as open under even the most abstract of English definitions, it is the word we use in math, and it's here to stay.

Answer 2

Its just a generalisation of the 'standard' topology on $\mathbb{R}$, where open intervals are indeed open. There are other 'weirder' topologies you can give to $\mathbb{R}$, which may not allow the open interval to be open, but they have similar properties to the 'standard' topology.

The point of defining these weird topologies is as counterexamples to the hard definitions soon to follow in your class (well, I found them hard). For example, this one answers 'Is the open unit interval on $\mathbb{R}$ with the topology generated by closed intervals open?' and the answer is yes, by Hurkyl's comment - this is the discrete topology.

You will need to get used to seemingly abstract objects being named after one example. Closed intervals, 'continuous' functions, 'boundary' of a set... The idea is to take our understanding on a simpler world (e.g. $\mathbb{R}$), define it rigourously, and see how far we can go with it.

Answer 3

The key idea you need, I think, is that a topological space is not a set of points.

You're misled by the fact that certain sets of points you usually see laid out in a particular way, and think their relationships to each other are a property of the set of points... but that's not true. The relationship between the points comes from how they're laid out. e.g. consider the rational numbers. You usually imagine them as being laid out on the rational number line:

(image taken from here)

(note the rational number line is not a continuum: e.g. there is a "hole" where $\sqrt{2}$ should be) And in this arrangement you visualize things like the sequence $1, 1/2, 1/4, 1/8, 1/16, \ldots$ converging to zero.

However, I could instead pick an enumeration of the rationals and arrange them like this:

$$ \begin{matrix} \ldots & \bullet^{-1/3} & \bullet^{-2/1} & \bullet^{-1/2} & \bullet^{-1} & \bullet^0 & \bullet^1 & \bullet^{1/2} & \bullet^{2/1} & \bullet^{1/3} & \bullet^{3/1} & \ldots \end{matrix} $$

Now, the sequence $1, 1/2, 1/4, 1/8, 1/16, \ldots$ rapidly shoots off to the right and doesn't converge to anything. We could even make this a metric space, insisting that if two rational numbers are separated by $n$ places, then the distance between them is $n$. e.g. in this arrangement, $d(-2, 1/2) = 5$.

Our intuition suggests a topology here too: the discrete topology. The points are all widely separated, so each point is contained in a "small" open set that contains nothing else. In fact, using the metric defined above, we can even define the open sets in the usual way: the open ball around a point $x$ of radius $r$ is the set of points $y$ with $d(x,y) < r$.

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The point to take away from this is that you need something additional to tell you how the points are "arranged". A topology is one sort of this additional information.

Note that topology doesn't tell you everything one might be interested in: e.g. it can't tell the difference between the curve | and the curve S; you need something else to distinguish between those. (e.g. the notion of a "curve in the Euclidean plane")

The intuition behind a topology is that a topological space is made up of open sets. In fact, some forms of topology (e.g. "pointless topology") go so far as to define the basic object (e.g. "locale") in a way that only talks about opens and makes no reference to the idea of a point at all. (the notion of point enters this theory in a much different way)

Now, even still, there are topological spaces where the open sets lack many of the nice properties we're used to from Euclidean geometry; but they still retain all of the properties essential to topology (e.g. finite intersections and arbitrary unions of opens are also open).