Elementary Topology Homework: Boundaries of (x,y) on y = |x-2| + 3 - x

In summary, we are trying to determine if the set of points (x,y) on y = |x-2| + 3 - x is bounded/unbounded, closed/open, connected/disconnected and what its boundary consists of. According to the definition of a closed set, it is closed and its boundary is itself. However, this is not always the case, as shown by the example of the unit 2-ball. As for boundedness, it appears to be unbounded when graphed. To determine connectedness, we need to consider if the function is continuous and if its graph has any breaks. The presence of an absolute value suggests that it may be disconnected, but further analysis is needed.
  • #1
tracedinair
50
0

Homework Statement



Determine if the set of points (x,y) on y = |x-2| + 3 - x are bounded/unbounded, closed/open, connected/disconnect and what it's boundary consist of.

Homework Equations



The Attempt at a Solution



I know that the set is closed, and then by definition of a closed set it's boundary is itself. As far as bounded/unbounded goes, it seems unbounded when I graph it because I cannot see the entire graph. I'm unsure about connectedness and do not know how to determine it.

Any help is appreciated.
 
Physics news on Phys.org
  • #2
Well, in (x,y) x is certainly not bounded. It runs from -infinity to infinity. To think about connectedness, do you see that the function is continuous? That means its graph is a continuous curve with no breaks in it, right? What's the definition of 'connected' that you are using?
 
  • #3
tracedinair said:
I know that the set is closed, and then by definition of a closed set it's boundary is itself.

Just so you know, that's not true in general. A closed set contains its boundary. For example, the unit 2-ball [tex](x^{2} + y^{2})^{1/2} \leq 1[/tex] is closed but is not the same as its boundary which is the 1-sphere [tex](x^{2} + y^{2})^{1/2} = 1[/tex]. I could just be being pedantic though and you could well have known that and just not felt like spelling it out.
 
  • #4
My guess then would that it is disconnected because of the absolute value in the function.
 
  • #5
tracedinair said:
My guess then would that it is disconnected because of the absolute value in the function.

Guess?? Why are you guessing?? I'll ask you once more. What's the definition of a connected set?
 

FAQ: Elementary Topology Homework: Boundaries of (x,y) on y = |x-2| + 3 - x

What is the purpose of studying elementary topology?

Elementary topology is the study of the properties and relationships of geometric objects, such as points, lines, and surfaces. It helps us understand the fundamental concepts of space and shape, and has many practical applications in fields such as mathematics, physics, and engineering.

What is the definition of a boundary in topology?

In topology, a boundary is the set of points that are on the edge or boundary of a shape or space. These points are not contained within the shape, but are essential in defining its boundary and distinguishing it from other shapes.

How do you find the boundary of a function in topology?

The boundary of a function on a given domain can be found by determining the set of points where the function is discontinuous or undefined. In the case of the function y = |x-2| + 3 - x, the boundary would be the set of points where x = 2.

What is the significance of the boundary in topology?

The boundary of a shape or space is important in topology because it helps define the shape and distinguish it from other shapes. It also plays a key role in determining the interior and exterior of a shape, as well as the connectivity and compactness of a space.

How does the boundary of y = |x-2| + 3 - x affect the graph of the function?

The boundary of this function, which is x = 2, is a vertical line that intersects the graph at a single point. This point serves as the "corner" or "break" in the graph, where the function changes direction and becomes discontinuous. It is important to consider this boundary when analyzing the behavior of the function and its relationship to other functions or shapes.

Similar threads

Prove that if any f:X-->Y is continuous, X is the discrete topology
Replies
23
Views
2K
Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
Replies
14
Views
2K
Path-connectedness for finite topological spaces
Replies
3
Views
2K
Continuity of a function under Euclidean topology
Replies
2
Views
2K
How to find the volume between y = 2√xz and y = 1?
Replies
28
Views
2K
Relation between components and path-components of ##X##
Replies
4
Views
2K
Are the following subsets open in the standard topology?
Replies
16
Views
2K
How do I correctly find the area bounded by x=-3, y=-x^2-2x, and y=x^2-4?
Replies
3
Views
1K
Determine all of the open sets in given product topology
Replies
3
Views
1K
Topology on a set ##X## (find interior, closure and boundary of sets)
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top