Topology, Connected and Path Connected

Connected

A set is connected if it cannot be partitioned into two nonempty subsets that are enclosed in disjoint open sets. Applying this definition to the entire space, the space is connected if it cannot be partitioned into two open sets. Equivalently, the only sets that are both open and closed are the entire space and the empty set.

If T is the image of S through a continuous function f, and two open sets separate two pieces of T, pull these back to find two open sets that separate S. If T is disconnected then so is S. Turning it around, the continuous image of a connected set is connected. A continuous function may blend components together, but it can't pull them apart.

Consider the closed unit interval from 0 to 1. Let this be the entire space, or embed it in n dimensional space if you prefer. If it is not connected then partition it into sets S and T. Assume, without loss of generality, that T contains 1. Let u be the upper bound of S. Now u is in S or in T. Suppose it is in S. Every open set containing S contains some of T, and S and T cannot be housed in disjoint open sets. similar trouble arises if u is in T, hence the unit interval is connected.

Path Connected

A path is the image of the closed interval [0,1] under a continuous function. Since the unit interval is connected, every path is connected.

If h is a path's function, the path connects x and y if h(0) = x and h(1) = y. Clearly x is connected to itself, and path connectivity is symmetric and transitive. This is an equivalence relation. All the points in a path connected component can be connected to each other. A set, or space, is path connected if it consists of one path connected component.

The continuous image of a path is another path; just compose the functions. The image of a path connected component is another path connected component. Again, a continuous function may blend path components together, but it cannot pull them apart.

If two spaces are homeomorphic, connected components, or path connected components correspond 1-1. The number of components and path components is a topological invariant. Deform the space in any continuous reversible manner and you still have the same number of "pieces".

Path Connected Implies Connected

Let C be a path connected component and suppose it is not connected. Separate C into two disjoint open sets and draw a path from a point in one set to a point in the other. Our path is now separated into two open sets. This contradicts the fact that every path is connected. Therefore path connected implies connected.

Connected but not Path Connected

Connected and path connected are not equivalent, as shown by the curve sin(1/x) on (0,1] union the origin. Note that values of x between 0 and 1 inclusive correspond 1-1 with points in this rather wild set. Disconnect the sin curve into two pieces A and B, where A houses the origin. Let t be the lower bound of the x coordinates of the points in B. If t is positive look at t,f(t) on the curve. Every open ball about this point includes points from A and B, so if t,f(t) belongs to either A or B, that set is not open. This is a contradiction, hence t = 0. Our set B includes x coordinates arbitrarily close to 0. Now A is open, so A includes a positive x coordinate. Let A include the x coordinate t, while B includes the x coordinate s, with s < t. Now A and B disconnect the curve from s to t, which is homeomorphic to a closed interval. A closed interval cannot be disconnected, hence our sin curve is connected.

Next suppose the set is path connected, thus there is a path from the origin to the point 1,sin(1), and this path is restricted to our curve. Let f be the continuous function that implements the path, with f(0) = 0. Let t be the lower bound of the values of x for which f(x) is not 0. Everything below t maps to 0, and by continuity f(t) equals 0. The path just sits at the origin for a while, then moves on. We may as well remove the segment from 0 to t. Reparameterize the path so that f(x) is nonzero for values of x arbitrarily close to 0.

Remember that f is continuous at 0. Choose a disk of radius ½ about the origin, and consider the preimage of our sine curve, as it intersects this disk. In other words, we want the preimage of points on the sine curve that are never farther than ½ from the origin. The preimage is open, a union of base open sets, and one of these base open sets is [0,δ). This is a connected half open interval, and its image under the continuous function f is connected. The image cannot contain more than one of the arcs of the sin curve slicing through the open disc of radius ½, for that would not be a connected set. Yet the image contains points away from the origin, so it contains one of the arcs of the sin curve. This arc, and the origin, form a disconnected set, and that is a contradiction. The set is not path connected.

Closure is Connected

Assume S is a connected set, and a larger set T is trapped between S and the closure of S. Try to split T into T₁ and T₂, contained in open sets O₁ and O₂. Intersect with S and find S₁ and S₂ in O₁ and O₂. But are S₁ and S₂ nonempty? If all of S is in O₁, then the complement of O₂ contains the closure of S, and none of T is in O₂. We did not split T after all. Thus S₁ and S₂ are nonempty, and S is disconnected, which is a contradiction. Everything between a connected set and its closure, including its closure, is connected. The open disk in the plane is connected, and so is the closed disk, including its boundary.

Overlapping Connected Sets

Let an arbitrary number of connected sets share a common point x. Let U be the union of these sets. Separate U into O₁ and O₂ and assume x is in O₁. Each connected set lies entirely in O₁, else it would be separated. Therefore all of U lies in O₁, and U is connected. A similar result holds for path connected sets. Draw a path from any point w in any set, to x, and on to any point y in any set.

Locally Connected and Locally Path Connected

A set S in a space T is locally connected if, for every point x in S, and every open set U containing x, there is an open set Q ⊆U containing x such that Q is connected. Note that Q∩S is actually the connected set, that is, Q is an open set in the subspace S, and is connected. Of course, if Q could be separated by open sets in our space, intersect these sets with S and Q could be separated inside S, hence Q is connected, period.

A similar definition exists for locally path connected. Every x,U has its open neighborhood Q in S such that Q is path connected.

With some imagination, you can build a connected set that is not locally connected. Let x be the pathological point. An open set containing x is disconnected, and x is in one of the two open components. this too is disconnected, and x is in one of the open components, and so on. So we're going to need a descending chain of open sets. Let U₁ U₂ U₃ etc be such a chain, intersecting in the point x. Each U_i is open, and each U_i-U_i+1 is also open. Now include a connecting point c. The only open set containing c is the entire space. Thus the space is connected, even though it is not locally connected at x.

A variation of the above builds a path connected space that is not locally path connected. Each U_i is an open region in [0,1), and c is a point in the plane below the real line. Each U_i+1 is the left "piece" of U_i, crunching down to 0. Draw an arc from c to each U_i. I'll leave the details to you.

Of course a set with two components could be locally [path] connected, but not [path] connected.

Some connected sets aren't even locally path connected. The earlier curve sin(1/x) union the origin is not locally path connected; the origin always causes trouble. Note, if it were locally path connected, it would be path connected, as shown by the next theorem.

Connected plus Locally Path Connected Implies Path Connected

Let C be a connected set that is also locally path connected. Pick any point x in C, and let U be the set of points in C that are path connected to x. Thus U is a subset of C.

Let y be a point in U. Enclose y in an open set H in C, such that y is path connected to all of H. Since an arc can run from x to y to anything in H, H is in U. Therefore U is the union of open sets and is open, relative to C.

Let y be a point in C that is a limit point of U. Put an open set H around y such that H is path connected. Let z be common to H and U. Now x connects to z connects to y, and y is in U.

Since U contains its limit points it is closed. thus U is both open and closed in C. If U is not all of C, separate U and the rest of C in open sets. This contradicts the fact that C is connected. Therefore U is all of C, and C is path connected.

In n dimensional space, every open ball is path connected, and every open set is locally path connected, hence every open connected set is path connected.

Product Space

Let P be the product of many spaces, using the standard (weak) product topology. We went to a lot of trouble to show that a function into P is continuous iff all its component functions are continuous. Now all that work pays off. Let the domain be the unit interval, and a path in P corresponds to paths in the component spaces. In other words P is path connected iff all its components are path connected.

If P is the product of finitely many spaces, it is locally path connected iff its components are locally path connected. Find path connected open sets in the components and put them together to build a path connected open set in P; or take the path connected base open set in P and find path connected open sets in the components.

Now ask whether P is connected. Suppose C, one of the component spaces, is not connected. Split C into open sets S and T and cross these sets with all other component spaces to partition P into two open sets, whence P is also disconnected. A similar argument applies when W, a subspace of P, has a projection that is disconnected in one of the components. Separate the projection in two open sets and cross with all other sets to find two disjoint open sets that separate W. If something casts two separate shadows, it must be in two pieces.

Conversely, assume all the component spaces are connected. I'm pretty sure P is connected, but I can't prove it right now. If you have a proof or counterexample please send it along.