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.
Picture1
(Plot of example 3. Try plotting other examples in Geogebra, zoom in and see why they are interesting.)

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.)

This article was imported from my old blog adding more information. Information is provided free of cost, and hoping it will benefit you. Please do point out if there are any mistakes/leave your comments below if you found it useful!

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