Let be a topological space. A path in is a continuous function . If and then we say that is a path from to and write .
This picture shows a path from the point to the point in the space :
Explicitly, the formula is
Note, however, that pictures like the one above can be a little misleading, because they only show the points traversed by the path, not the time at which those points are reached. Consider the maps given by and . The resulting tracks are just the same:
However, if we add markers for then we can see the difference:
Suppose that is a subset of , with the subspace topology. For any points , we can define by . This gives a path from to in , which we call a straight line path. However, this path might or might not lie in ; in any case where we want to use straight line paths, we need to check this. For example, if is a circle then the straight line path from to is not contained in , except in the trivial case where . In the space shown on the right below, the straight line paths from to and from to are contained in , but the straight line path from to is not contained in .
Later, we will want to consider paths as continuous maps rather than continuous maps . We will always identify the point with the point . This ensures that the point gets identified with , and the point gets identified with .
Let be a topological space.
For any , the constant path is just given by for all . This is a path .
Now suppose we have a path . We define a path by , and we call this the reverse of .
Now suppose we also have a path . We define a path by
(Roughly speaking, in the first half-second we go from to by following at double speed, then in the next half-second we go from to by following at double speed.)
There are a number of things that we need to check in order to validate the above definitions, especially part (c). Firstly, is continuous by Example 3.22, and is continuous by Proposition 3.24, because it is the composite of with the continuous function .
Now consider . Firstly, if then both clauses in the definition of are applicable. This would be a problem if the two clauses gave different answers for the value of . However, the first clause gives the answer , and the second clause gives the answer . As and we have so the two answers are the same.
Next, we need to show that the map is continuous. We can write as , where and . By closed patching (Proposition 3.35), it will be enough to show that is continuous on and also continuous on . On , we have . This is the composite of with the map , and both of these maps are continuous, so is continuous by Proposition 3.24. A similar argument shows that is continuous on , so it is continuous on as required. We also have and , so .
We introduce a relation on by declaring that iff there exists a path in .
This relation is an equivalence relation.
(If you need to review the basic ideas about equivalence relations, you can look forward to Definition 7.13.)
We must show that the relation is reflexive, symmetric and transitive. In more detail, the conditions are as follows:
For all , we must have .
For all with , we must have .
For all with and we must have .
For condition (a), we always have , and this shows that . For condition (b), suppose that . This means that there exists a path . It follows that the reverse path has , and thus that . Finally, suppose that and . This means that there exist paths and . The joined path then goes from to , proving that as required. ∎
The equivalence classes for are called the path components of . We write for the path component containing , so that iff . We write for the quotient set , or equivalently, the set of path components. We say that is path connected if it has precisely one path component. This means that , and for all .
Let be a subset of . We say that is convex if for all and we have . Equivalently, this means that any straight line path with endpoints in is contained wholly in , and so counts as a path in between those endpoints. It therefore follows that is path connected (provided that it is not empty).
In particular, the ball , the cube and the simplex are all nonempty and convex, so they are path connected.
Consider again the following space , which we discussed in Section 2:
This is the disjoint union of the three subsets , and . The set is the set of all points that can be connected to by a continuous path in , or in other words . Because , and all lie in they can all be connected to each other, which means that and . In the same way, we have and . From this we see that and so .
You should not be confused by the fact that itself is an infinite set. The whole set taken as a single object is an element of , the whole set taken as a single object is another element, and the whole set is the third element.
For , the sphere is path connected.
Suppose we have points . Suppose for the moment that they are not opposite points, so . Consider the linear path given by . Because and are not opposite, we see that the straight line from to does not pass through the origin, so is never zero. It is therefore legitimate to define . This gives a continuous map with and , so in . Now consider the exceptional case where . If we allowed the case then would just consist of two points, but we have specified that , so is infinite, so we can choose a point that is different from and . The ordinary case now tells us that and , and is an equivalence relation so as required.
∎
Let and be points in a topological space . Suppose that there is a continuous function such that
for all .
.
Then .
Suppose (for a contradiction) that there is a path from to in . We then have a continuous map with and . By the Intermediate Value Theorem, there exists with . This means that the point satisfies , contradicting assumption (a). We therefore conclude that no such path can exist, so . ∎
Consider the space . If with , we can define by . This is nonzero for all , and satisfies , so . It follows that the connected components are just the singleton sets for all , so is essentially the same as .
Consider the space of invertible matrices, and the elements , where
We can define a continuous map by . It is a standard fact of linear algebra that invertible matrices have nonzero determinant, so is nonzero everywhere on . We also have and . Proposition 5.15 therefore tells us that . More generally, we could take any and put
The same line of argument shows that if and then . However, it can also be shown that the subsets and are both path connected (we will not give the proof here). Assuming this, we see that and are precisely the path components of , so and .
Suppose that can be written as , where and are open subsets of with . Then for and we have .
We define by for all and for all . (This defines for all , because . There is no clash between the two clauses, because .) We claim that is continuous. (Assuming this, the main claim follows by Proposition 5.15.) Consider an open subset ; we must show that is open in . We have
In all cases, we see that is open, as required. ∎
Let be a continuous map between topological spaces. Suppose that with , so that in . Then we also have , and so in .
We are assuming that , which means that there exists a continuous function with and . Put , so is a continuous function from to . It has and , so . As there exists a path in from to , we have as claimed. ∎
Let be a continuous map between topological spaces. Then there is a well-defined map , given by for all . Moreover:
For the identity map , we have .
For any pair of continuous maps and , we have .
Let be an element of ; we need to define . We can choose a point such that , and we want to define . The only problem with this is that it seems to depend on the choice of . If we chose a different element with , then we would also want to define , and that would be inconsistent if was different from . However, Lemma 5.19 tells us that , so this problem does not arise, and we have a well-defined function as claimed. We also have , so is the identity map. Similarly, if is another continuous map, we have
so . ∎