Topology is the study of the properties of shapes which are unaffected by smoothly stretching or squeezing or twisting the shape (but not breaking or tearing it).
So a topologist is a mathematician who can’t see the difference between a ring doughnut and a coffee cup.
To see the difference between tearing and smoothly stretching or squeezing or twisting, we have to have some idea of continuity. The conceptual difference here is similar to the difference between a continuous curve and a discontinuous one. And we need some idea of the edges of shapes, which requires the concept of convergence.
We can study “topological equivalence” of shapes in the three-dimensional Euclidean space, , in which we’re thinking of the doughnut and coffee cup. But also in more general mathematical contexts.
In those more general contexts, we need to know what structure distinguishes between stretching or squeezing or twisting, and tearing, or, in other words, defines continuity and convergence.
Topologists work with metric spaces, in which a metric, a concept of distance, is defined. That allows us to see what is close and what is distant, what has moved smoothly and what has “jumped” or torn apart.
When they want more generality, they work with topological spaces. We say that a topology is defined on a space when we have workable rules to define in it what is:
- an open set (something analogous to the interval containing all numbers greater than 0 and less than 1, written ), which has the property that every point in it has a surrounding neighbourhood (sub-interval), maybe a small one but a neighbourhood, completely contained within the interval)
- a closed set (something analogous to the interval containing all numbers greater than or equal to 0 and less than or equal to 1, written ), which has the property that every point which is a limit of a sequence of points in the interval is also within the interval.
is not closed, because the sequence has a limit, 0, outside the interval.
is not open, because every neighbourhood (sub-interval) completing surrounding 1 contains some points > 1 and so outside the interval.
You can then show that a subset A of a metric space X is closed if and only if its complement X−A (meaning: all the points in X which are not in A) is open.
A topology on a space X is a set of rules for defining “open” subsets for X which fits these conditions:
- (the subset with nothing in it) is open
- X is open
- if you stick together any amount of open subsets, the “total” subset (containing all the points which are in any of those subsets) is open
- if you have two open subsets A and B, and C is the subset of points which are in both A and B, then C is open.
At one extreme, rules which say that only and X are open define a topology.
At the other extreme, rules which say that every subset of X is open also define a topology.
Usually we want something in between.
For example, if X is an infinite set, the set of rules which says that A is open if A is or X−A is finite (i.e. if A contains all but a finite number of the points in X) is called the Zariski topology on X.
Relative to a topology, we can decide whether a space is connected (all in one piece):
A space X is connected if the only subsets of X which are both open and closed are and X.
Then a transformation or mapping f from X to Y is continuous if, for every open subset B in Y, the subset of all points in X which f maps into B is also open.
f is a homeomorphism (it defines X and Y as shapes which can be stretched, squeezed, or twisted into each other, without tearing) if f is a continuous one-to-one mapping, and its inverse (what you get by backtracking it) is also continuous.
For much more, presented clearly and in manageable chunks, see
http://www-history.mcs.st-and.ac.uk/~john/MT4522/index.html, which I’ve used in compiling this summary.
As those notes point out, topology has developed in different directions:
1. Differential topology studies surfaces, solutions of differential equations, etc.
2. Algebraic topology: the study of algebraic (in the university sense: groups, rings, etc.) invariants of topological spaces.
3. Combinatorial or geometric topology
4. General or point-set topology is the basic theory underlying the above.