Intuitive approach to topology

You might first want to study analysis, which will give you more of a motivation for learning topology. Analysis introduces you to many concepts in topology in a more tangible way, in more familiar contexts like the set of real numbers and metric spaces, where you at least have a notion of distance.

After analysis, you could study topology equipped with better intuition. This is the usual progression at the college level as well.

As for a text for introductory analysis, I recommended Principles of Mathematical Analysis by Walter Rudin. Chapter 2 covers the basic ideas of topology relevant to analysis.

The other answers are good, but appeal to ideas that might not be familiar to a high school student (metric spaces), or to things which I feel are an important consequence of topology but not a motivation/intuition (continuity, which requires extra ideas, like maps between topological spaces).

A topology is one of the weakest structures you can put on top of a bare set of objects. Very roughly, (my intuition is) it allows one to say which elements of the set are 'nearby' each other. It does so in rather an ingenious way - one which does not need to appeal to measurement or distance. Instead it uses the idea of open sets, which you may or may not be familiar with. One specifies 'a topology' by specifying its open sets, which are subsets with particular properties under unions and intersections. There are usually many possible distinct topologies available for any one set.

The open sets which contain a particular element from the big set are often called the 'neighbouroods' of that element. Sometimes one neighbourhood can lie entirely inside another (via set inclusion), so is a 'smaller' neighbourhood. Hence the intuition: one can think of the elements of a smaller neighbourhood as being 'closer' than those of a larger one.

This is then where we can talk about the ideas of metrics, which measure distance, or continuity, which requires one to talk about the relative 'sizes' of sets in the domain and image of a function. It's amazing that even with something so simple as a topology one can already talk about such powerful analytic ideas. But topologies on their own are so dreadfully unconstrained that some very strange structures are possible. Usually, at least in my line of work, a topology is used in conjunction with some other structure in which that intuition is realised in a precise sense. Then what the others have said - about continuity, etc - comes into play, where the topology does most of the low-level analytic heavy lifting.

We'd like to say:

A topological space is a set $X$ together with a notion of convergence relating sequences in $X$ with points of $X$. Given a sequence $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every sequence converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all sequences $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

Unfortunately, this is completely wrong. For starters, you can't always recover the topology just from knowing which sequences converge to which points. Though sometimes you can; these are called sequential topological spaces. But in general, what your convergent sequence/point pairs isn't enough to determine the topology, and we have to pass to "generalized sequences"; the technical term is net. So, we want to say:

A topological space is a set $X$ together with a notion of convergence relating nets in $X$ with points of $X$. Given a net $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every net converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all nets $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

This still doesn't quite work, because the "set" of all nets in a topological spaces turns out to be too big to form a set; they merely form a class. This creates some technical problems that we'd rather do without. The usual workaround goes like so:

• Given a set $X$, there's a notion of filter in a set.
• Every net corresponds to something called its "eventuality fiter."
• We can decide whether or not a net $x$ converges to a point $y$ just from knowing the eventuality filter of $x$.
• Hence instead of equipping $X$ with data regarding which net/point pairs are convergent, the usual workaround is to equip $X$ with data regarding which filter/point pairs are convergent, and treat convergence of nets as a derivative notion.

So, our definition becomes:

A topological space is a set $X$ together with a notion of convergence relating filters in $X$ with points of $X$. Given a filter $x$ and a point $y$, we can ask whether or not $x$ converges to $y$, and certain axioms hold regarding this relation.

A topological space is called Hausdorff iff every filter converges to at most one point.

A function $f : A \rightarrow B$ between topological spaces is continuous iff for all filters $x$ in $A$ and all points $y \in A$, we have that if $x$ converges to $y$ in $A$, then $f(x)$ converges to $f(y)$ in $B$.

There's still a subtle issue. If you try to axiomatize topological spaces via filter convergence, you wind up with a significantly more general notion called a convergence space. Convergence spaces are really nice, and personally I wish people would start treating them as the basic objects of interest in general topology, and treat topological spaces as a mere special case. Unfortunately, I haven't been able to find an elementary introduction to such things that I can link you to, so you'll have to learn things the classical way until you're ready to go off on your own.