# Analysis/Topology: Connected Sets in Metric Spaces

Here’s a small discussion on definition, examples of connected sets in Metric Spaces.

## Definitions

• Definition 1: Let <M,ρ> be a metric space and let A be a subset of M. Suppose <A,ρ> is itself regarded as a metric space, then if there is no set except A and ∅ which is both open and closed in <A,ρ>, we say that A is connected.
• Definition 2: Let <M,ρ> be a metric space and let A be a subset of M. When <A,ρ> is itself regarded as a metric space, then if there is no set except A and ∅ which is both open and closed in <A,ρ>, we say A is connected.
• Definition 3: Let M be a metric space. Then M is connected if and only if every continuous characteristic function on M is constant.
• Definition 4: (In $\mathbb{R}$) The subset A of $\mathbb{R}$ is connected if and only if a∈A,b∈A with a<b, then c∈A for any c such that a<c<b.

## Theorems/Facts/Properties

• Let f be a continuous function from a metric space $M_1$ into a metric space $M_2$. If $M_1$ (the domain of f) is connected, then the range of f is also connected.
• If f is a continuous real valued function on the closed interval [a,b], then f takes on every value between f(a) and f(b). (Intermediate Value Theorem)
• If $A_1,A_2$ are connected subsets of a metric space, and if $A_1\cap A_2$≠∅, then $A_1\cup A_2$ is also connected.

## Examples

1. (In $\mathbb{R}$) Any interval is connected in $\mathbb{R}$. For example, $[0,1],(10,20]$
2. (In $\mathbb{R}^2$)  Define B = { (x,y) ∈$\mathbb{R}^2$ : y = sin⁡ $1 \over x$, 0 < x $\le$ 1} $\cup${(0,y)∈ $\mathbb{R}^2$:-1<y<1}. Then B is Connected in $\mathbb{R}^2$.
3. (In $\mathbb{R}^2$)  Define C = {(x,y)∈$\mathbb{R}^2$ : y = x sin⁡$\frac{1}{\sqrt{x}}$,0<x $\le$ 1} $\cup$ {(x,y) ∈ $\mathbb{R}^2 : (x+1)^2+y^2\le 1$}. Then C is Connected in $\mathbb{R}^2$.
4. (In $\mathbb{R}^2$)  Define B = { (x,y) ∈$\mathbb{R}^2$ : y = x sin⁡ $1 \over x$, y=1 when x=1} is connected.

## Counterexamples

1. (In $\mathbb{R}$) $[0,1]\cup (6,20]$ (Union of two disjoint intervals)
2. (In $\mathbb{R}^2$)  Define B = { (x, y) ∈ $\mathbb{R}^2$ : y = x sin⁡ $1 \over x$, -1 < x $\le$ 1} is not connected. (It is undefined at x = 0)
3. (In $\mathbb{R}^2$)  Define C = {(x,y)∈$\mathbb{R}^2$ : y = x sin⁡$\frac{1}{\sqrt{x}}$,0<x<1} $\cup$ {(x,y) ∈ $\mathbb{R}^2 : (x+1)^2+y^2\le 1$}. (Compare it with third example above – what changes?)

You can find more example and counterexamples here (link).

## Quizzes

Consider the sets , then
A is connected, B and C are not connected
A and B are connected, but C is not connected
B and C are connected, A is not connected
A and C are connected, B is not connected
Let , then
The range of f is connected
The domain of f is connected
The co-domain of f is conected
Note that the domain of f is connected. Hence, range is connected. But the codomain is not connected (union of disjoint intervals).
None of the above
Note that the domain of f is connected. Hence, range is connected. But the codomain is not connected (union of disjoint intervals).
Let , then
A is connected in
A is not connected in (Try plotting the graph)
A is connected in
A U {(0, 1)} is connected in
All of the above
A is not connected in (Try plotting the graph)

(This is called the cardinal sine or sinc function. You can read more about it here (link). This is a very important function with regard to CSIR-NET.)