What Does Connected Really Mean in Different Mathematical Contexts?

[CMoore]

Defining the term "connected"

Some authors use the word "connected" as applied to a set S to mean that for any two points A and B in S there exits a (piecewise) smooth path in S from A to B. This is, for instance, the approach taken by Apostol in Calculus Vol II (p 332) and the approach taken by C.H. Edwards in Advanced Calculus of Several Variables (p 84)

On the other hand, topology texts generally define the term "connected" to just mean that S cannot be expressed as two disjoint nonempty subsets and use the term "path connected" to indicate that any two points can be joined by a path (i.e., a continuous map from a closed interval into S) that lies entirely in S. (E.G, Munkres)

So, then, what Apostol and Edwards mean is that S is "differentiably path connected" or "smoothly path connected". Is there established (updated?) terminology that identifies a space as "differentiably path connected" in the sense that these authors intend? In other words, a space in which any two points can be joined by a smooth path that lies in the space is called ______ ?

Thanks.

 

[Tac-Tics]

The two kinds of connectedness are usually referred to as "simply connected" (only two unique open-closed subsets) and "path connected" (existence of a parametrized map). Path connectedness is a subset of simply connectedness.

Since the author is talking about differentiable *path* connectedness, it's the latter.

 

[quasar987]

The two kinds of connectedness are usually referred to as "simply connected" (only two unique open-closed subsets) and "path connected" (existence of a parametrized map). Path connectedness is a subset of simply connectedness.

Since the author is talking about differentiable *path* connectedness, it's the latter.

"Simple connectedness" is something different from "connectedness".

A space is said to be simply connected if it is path connected and if every closed curve can be continuously shrunk into a point.

A space is said to be connected if, like the OP said, it cannot be expressed as the reunion of two disjoint nonempty subsets. While in a connected space, the only clopen sets are the void and the space itself, this property does not characterize connectedness. Consider for instance the space consisting of three distinct points, X={a,b,c}, with the topology {X, void, {a,b}, {c}}. Here, {a,b} and {c} are clopen but X is not connected since we can write X={a,b} u {c}.

 

[quasar987]

CMoore, I don't know if there is an established convention. But probably, there is not, and for the following reason.

The context in which it makes sense to ask whether a space is "piecewise smooth path connected" is that of differentiable manifolds. I.e. topological spaces on which there is an additional structure designed to make sense of the notion of differentiability. And it is easy to see that all path connected smooth manifolds are in fact "piecewise smooth path connected". (And even, every connected component of a smooth manifold is "piecewise smooth path connected")

So the term "piecewise smooth path connected" is useless: as soon as it makes sense to talk about differentiability of a topological space, the notions of connectedness, path connectedness and piecewise smooth path connected coincide.

 

[Tac-Tics]

Ah, I apologize for my error in terminology. I guess it's not "simply connected", but simply "connected" =-)

 

[d_leet]

The two kinds of connectedness are usually referred to as "simply connected" (only two unique open-closed subsets) and "path connected" (existence of a parametrized map). Path connectedness is a subset of simply connectedness.

Since the author is talking about differentiable *path* connectedness, it's the latter.

Simply connected is by far a more specific condition than either of which you mention. Simply connected means path connected, and a trivial fundamental group. No representation as the disjoint union of two open subsets is referred as Connectedness, and if there exists a path between any two points in the space then that is referred to as path-connected.

 

[wofsy]

topological connectedness: the space is not the disjoint union of two non-empty open sets.

path connectedness: Any two points are joined by a continuous map from an interval into the space. No differentiability is required. In fact in many path connected spaces the idea differentiability is not defined.

In a space where differentiability is defined any continuous path may be uniformly approximated by a differentiable path. Apostol wants to avoid proving this because he only cares about piece wise differentiable examples. But his definition is ad hoc and not standard.

famous example: In the Cartesian plane,the graph of sin(1/x) for x not zero united with with the vertical interval on the y-axis lying between -1 and 1 is connected in the subspace topology but not path connected.

another example: The graph of a path generated by a continuous Brownian motion. The curve is path connected (since it connects itself) but is nowhere differentiable so there is no piece wise smooth path connecting any two of its points.

problem: Assume that the Continuum Hypothesis is true and select a total ordering of the numbers between zero and one. Consider those points in the unit square in the xy - plane whose y co-ordinate is greater that its x co-ordinate in this ordering. Is this space connected?