Explaining what is Pathwise-connectedness.

Question

I'm an average guy but interested in explaining myself maths through illustrations and intuition(which at times fails!!!).I'm preparing myself for my calculus of several variables exam.

I'm studying connected sets and path-wise connected sets.

A topological space $X$ is pathwise-connected iff for every two points $x,y$ in $X$, there is a continuous function $f$ from $[0,1]$ to $X$ such that $f(0)=x$ and $f(1)=y$. Roughly speaking, a space $X$ is pathwise-connected if, for every two points in X, there is a path connecting them.

I can't understand this,why such a function $f$ has been created.Is there some nice intutive,geometrical way to understand what is path-connectedness?Please help....

Answer 1

The important word was continuous.

Roughly speaking, a function $f$ from $[0,1]$ to $\mathbb{R}$ is continuous if and only if you can draw its values with pencil, without lifting it from $X = 0$ to $X = 1$. When you do this, the figure you create looks like a path.

Here you can see an example of a continuous function (left), and a discontinuous function (right).

As you can see, the second picture doesn't represent a path, because there is a "hole".

The definition you gave use a continuous function to formally define the "path".

So a space $X$ is space-connected if for each couple of points, you can draw a line between them. For example a disc is space connected, but the union of two disjoint circles isn't (because we can take the first point in one circle and the second in the other).

Note that the pencil analogy still holds when $X$ isn't $\mathbb{R}$, just imagine going it with a pencil that can write in the air. Or even simplier, create a path with your finger without exiting $X$.

http://en.wikipedia.org/wiki/Connected_space#Path_connectedness

Answer 2

Imagine Bob the bug walking from $0$ to $1$ on the real number line. When Bob is at $x\in [0,1]$, Bob's sister bug, Brenda is at $f(x)$ in $X$. Because $f$ is continuous, Brenda traces out a path in $X$ from $f(0)$ to $f(1)$ while Bob goes from $0$ to $1$ in $[0,1]$.

So $f(0)$ and $f(1)$ are connected by a path.

Answer 3

You can always think of a path as a journey. It is common in texts to define a path as you give, which you can think of as of "time" $1$, but some texts, for example this one, define a path as a map $f:[0,r] \to X$ which can be thought of as a journey of time $r \geqslant 0$ from $f(0)$ to $f(r)$. If also $g:[0,s] \to X$ is a path and $f(r)=g(0)$ then you get a path of "time" $r+s$ from $f(0)$ to $g(s)$. I'll leave that to you to work out. This "composition" of "journeys" is associative. A path of time $0$ means you go nowhere, and you can also do this for time $r> 0$, which is known as "waiting".

One does want a journey to be continuous; we do not usually have a "beam me up, Scotty".

An island is path connected, but an archipelago is not.

Answer 4

Here, $$f(t), \quad t \in [0, 1]$$ is a parameterization of a curve, that is, it's an instruction for how to draw a curve on the space $X$: At each time $t$, the pencil is at the point $f(t)$ in $X$. Continuous means that you don't have to lift your pencil up when following the instructions, that is, that the pencil doesn't suddenly "jump" from one point to a faraway point.

Of special interest are the starting point $f(0)$ and the ending point $f(1)$ of a curve. We say that $X$ is path-connected if any two points in $X$ can be connected by a curve without lifting your pencil (or otherwise leaving $X$).