filtersintuition
In "Counter-examples in topology" of Steen and Seebach, they define a filter on a set $X$ is a collection F of subsets of $X$ with the following properties:
Every subset of $X$ which contains a set of $F$ belongs to $F$
Every finite intersection of sets of $F$ belongs to $F$
$\varnothing$ is not in $F$
My question is: what is the intuition behind this definition?
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.
Here are some general pointers for gaining intuition in topology:
Learn lots of examples early, and use them to guide your understanding. Take the definition of a topology, for instance. The original motivation for this definition comes from familiar topological spaces, such as the real numbers or, more generally, $\mathbb R^n$ or, more generally still, metric spaces. Learn the definition of continuous maps $\mathbb R^m\to \mathbb R^n$ and between metric spaces in general and then try to understand why the 'open-set' definition of a continuous map is equivalent to these definitions. At this point, it is natural to ask how far we can generalize this definition. Topology is really about replacing '$\epsilon$-balls' in metric spaces with more general 'basic open neighbourhoods' that have precisely the right properties that let us make sense of concepts like 'continuous map' or 'small neighbourhood' that we're used to from real analysis.
Learn how to draw pictures of topological spaces. A seasoned topologist will naturally draw the diagram
for an open set $U$ contained in a topological space $X$ or
for the product space $X\times Y$, even if the spaces don't actually look like that at all. Drawing a diagram will help you get intuition for writing actual proofs.
Recognize that sometimes you're just going to have to do a lot of topology. The definition of a compact topological space is notoriously difficult for newcomers to the subject to internalize. Looking back at my own mathematical development, I don't think that there's anything that anyone could have said to me at the time that would have helped me understand it better. After a few years of using this definition, and working with compact spaces - primarily the closed interval $[0,1]$, I've got a much better understanding of its importance. Indeed, my advice to anyone who wants to understand compactness is to learn a proof of the Heine-Borel theorem (i.e., $[0,1]$ is compact) and then go back and write new proofs of the following theorems from real analysis using only the fact that $[0,1]$ is a compact topological space:
Keep asking (yourself) questions. You're doing exactly the right thing by questioning the definitions you're seeing. If you blindly accept every definition you're taught without questioning whether it's natural or 'correct' then you'll find that you get to the end of a course in topology without really understanding anything you've been taught. Every time you see a definition, ask yourself 'Why has it been defined this way?' At the same time...
Understand that there's no hurry. If you don't immediately get why a definition is used, don't worry about it and try and move on. Topology has been gone over and refined many times in the hundred or so years it's been around and you can be sure that the definitions you are seeing are natural and very useful. You can also be sure that, over the course of your studies in mathematics, you will come across topology again and again, and this will give you a chance to keep re-asking the questions you've been asking yourself. Each time you'll get more and more answers.
(optional) Learn equivalent definitions. The reason this is optional is that plenty of people have survived perfectly well with the usual definitions from topology, and you certainly shouldn't spend your time learning lots of other definitions if you're having trouble with the existing ones. However, it can be useful to learn some alternative ways of looking at things. For example, here are some alternatives to the 'open set' definition of a topology:
System of basic open neighbourhoods - This is pretty similar to the 'open set definition', but rather than requiring that our collection of sets be closed under unions and all finite intersections, we only require that the intersection of two basic open neighbourhoods can be written as the union of basic open neighbourhoods. This definition is technically less precise than the usual definition, since different systems of b.o.n.s can give rise to the same topology, but it is sometimes more natural. For example, the topology on a metric space is induced by the system of basic open neighbourhoods given by $\epsilon$-balls $B(x,\epsilon)$.
Closure operator It's interesting that the topology on a space $X$ can be deduced from the closure operator $\text{Cl}\colon\mathcal P(X)\to\mathcal P(X)$. This gives rise to an equivalent definition of a topological space; see Wikipedia.
For compactness:
Sequential compactness. This only works for metric spaces, but is still fun. A metric space is compact if and only if it is sequentially compact - every sequence has a convergent subsequence
Categorical compactness. A topological space $X$ is compact if and only if for every topological space $Y$ the projection map $\pi_Y\colon X\times Y\to Y$ is a closed map. See this note.
Best Answer
Generally each filter encodes a possible meaning of what it means for a subset of $X$ to be "large enough" or "contain enough points" for something.
In addition to Asaf's examples, other good filters to keep in mind are
The principal filter of $x\in X$, which is the set of all subsets of $X$ that contain $x$. Here "large enough" is almost trivial, being large enough just means to contain $x$.
The neighborhood filter of $x$, when $X$ is a topological space. This is the set of all subsets of $X$ that have $x$ as an interior point.
These example show that being "large" or (in Asaf's words) including "most" elements can be very context dependent and doesn't need to correspond to "most" in an objective sense.
The neighborhood filter is particularly important because it is the prototype of the use of filters to generalize the notion of limits. Filters are one possible answer to what kind of thing does "$x\to 5$" in $\lim\limits_{x\to 5} f(x)$ stand for.
For example, when $\mathcal F$ is a filter in $X$ and $f$ is some function defined on $X$, we can define $$\lim_{x\to\mathcal F} f(x)=y \text{ means that } \forall \varepsilon>0\, \exists F\in\mathcal F\, \forall x\in F : |f(x)-y|<\varepsilon$$ If we choose $\mathcal F$ to be the punctured-neighborhood filter at $5$ gives us the usual limit for $x\to 5$. But there are other filters that give one-sided limits ($x\to 5^+$ corresponds to the filter of all sets that contain $(5,5+\delta]$ for some $\delta>0$) or limits involving infinity ($x\to-\infty$ corresponds to the filter of all sets that contain $(-\infty,y]$ for some $y\in\mathbb R$), and so forth, which can now be considered instances of a single uniform concept.