R^2 accumulation and open/closed

  • Context: MHB 
  • Thread starter Thread starter Dustinsfl
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on the set of points \((x,y)\) defined by the inequality \(x^2 - y^2 < 1\), which is identified as an open set in \(\mathbb{R}^2\). It is concluded that all points on the hyperbola defined by \(x^2 - y^2 = 1\) serve as accumulation points. Therefore, the complete set of accumulation points is described as \(\{(x,y) \in \mathbb{R}^2: x^2 - y^2 \le 1\}\).

PREREQUISITES
  • Understanding of open and closed sets in topology
  • Familiarity with hyperbolas and their equations
  • Knowledge of accumulation points in metric spaces
  • Basic concepts of \(\mathbb{R}^2\) coordinate geometry
NEXT STEPS
  • Study the properties of open and closed sets in topology
  • Explore the characteristics of hyperbolas and their applications
  • Research accumulation points and limit points in metric spaces
  • Examine the implications of inequalities in \(\mathbb{R}^2\)
USEFUL FOR

Mathematicians, students studying topology, and anyone interested in the properties of geometric sets in \(\mathbb{R}^2\).

Dustinsfl
Messages
2,217
Reaction score
5
All points $(x,y)$ such that $x^2 - y^2 < 1$.

This set is open but I am not sure about the accumulation points.
 
Physics news on Phys.org
dwsmith said:
All points $(x,y)$ such that $x^2 - y^2 < 1$.

This set is open but I am not sure about the accumulation points.

The set is the region of \(\mathbb{R}^2\) between the branches of the hyperbola \(x^2-y^2=1\). All the points on the hyperbola are accumulation points, so the set of accumulation points is \( \{(x,y)\in \mathbb{R}^2: x^2-y^2\le 1 \}\)

CB
 

Similar threads

Undergrad Discrete Subrings of Complex Numbers: Topology of Rings
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
1K
Graduate Regarding fibrations between smooth manifolds
  • · Replies 1 ·
Replies
1
Views
2K
High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad About the definition of topological manifold using closed sets
  • · Replies 14 ·
Replies
14
Views
7K
Undergrad On two definitions of locally compact
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Proving a function f is continuous given A U B = X
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad On Borel sets of the extended reals
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Another similar notion to compactness
  • · Replies 2 ·
Replies
2
Views
397
High School Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?
  • · Replies 6 ·
Replies
6
Views
3K