Motivation behind topology

Solution 1:

Topology can mean different things in mathematics, depending on the context.

If a mathematician describes themselves as a topologist, this likely means that they study various kinds of shapes (technically, manifolds, or related kinds of spaces), with an eye perhaps to classifying them (a typical example being the classification of closed orientable surfaces via the number of handles attached to a sphere), or understanding certain invariants or other aspects of their structure (e.g. the Poincare conjecture, which gives a characterization of the three-dimensional sphere in terms of a certain invariant, namely its fundamental group).

On the other hand, the basic axioms of topology that you are asking about (a topology on a space is a collection of subsets, called open subsets, satisfying the following axioms ...) are much more general. There is a branch of mathematics that focuses on studying (more or less) these axioms and their consequences (appropriately known as general topology, where general refers to the generality of the axioms), but this investigation is rather different in flavour to the mathematics described in the preceding paragraph --- it is perhaps closer to set theory than it is to the investigation of the topology of manifolds and so on.

But thinking in terms of the subject of general topology is not a good way to understand the point of the axioms of topology. By analogy, consider the group axioms: there is a well-developed area of mathematics called group theory which studies (more or less) these axioms and their consequences, but the notion of group is ubiquitous in mathematics, and plays a fundamental role in many areas of mathematics besides group theory proper (including number theory, topology, and geometry). Thus the group axioms capture a concept which is of fundamental mathematical importance, and this is why we isolate the notion of group and study it.

Similary, the axioms of a topology, and the basic definitions associated to them (such as continuity of maps, and connectedness and compactness of subsets) were formulated as the consequence of a rather long attempt during the 19th and early 20th centuries to isolate and abstract the basic notions related to continuity and limits. These concepts are again ubiquitous in mathematics, and so the notion of topology (in the sense of the axioms you asked about) are applied in an enormous number of different contexts throughout mathematics (not just in those particular areas studied by topologists or general topologists, but in geometry, analysis, number theory, parts of algebra, logic, ...). Although the axioms may seem unusual, and unrelated to the notions of shape and position that you see discussed in an intuitive account of topology, they are in fact carefully crafted to capture, in very general language, the notions of continuity, nearness, limits, and so on.

In order to see the truth of my last claim, you will have to invest some time studying the axioms (i.e. learning some basic general topology), and then see how it is applied in various contexts. At least in the U.S., this typically happens in advanced undergraduate and beginning graduate courses.

Solution 2:

This math.SE question may be relevant, but not pedagogically optimal.

Pedagogically I think the simplest answer is to axiomatize topological spaces via the Kuratowski closure axioms. Instead of specifying what properties open sets or closed sets satisfy, the Kuratowski closure axioms specify a closure operator $S \mapsto \text{cl}(S)$ on subsets $S$ of a set $X$ and axioms it ought to satisfy. You should think of closure as axiomatizing abstract limits (that is, $\text{cl}(S)$ roughly corresponds to the set of all possible limits of sequences of elements of $S$, but it actually doesn't; we need to replace sequences by filters or nets in full generality). Then the closed sets are precisely the ones for which $S = \text{cl}(S)$ and the open sets are the complements of the closed sets as usual.

It's worth mentioning that the notion of a topological space is absurdly general. For many applications you'll only need to think about the topology of much more restricted types of spaces (e.g. manifolds, CW-complexes...). Nevertheless, because it is so general, it can be fruitfully applied to many areas of mathematics, and because the set of axioms used is relatively sparse, proofs in full generality are generally relatively short.

Solution 3:

This will answer only the question in the second paragraph.

Imagine the half-open interval $[0,1)$ with the usual open sets. In particular, an open neighborhood of $0$ contains $0$ itself and all positive numbers sufficiently close to $0$.

Now alter the definition of "open set", so that every open neighborhood of $0$ contains not only $0$ and all positive numbers close enough to $0$, but also all numbers close enough to $1$. I.e. $[0,\varepsilon)\cup(1-\eta,1)$.

Can you see how that alters the way in which the whole space is connected together?

Thus: how the space is connected together, is simply a matter of which sets are open. That's the connection between the two things.