Question

Prove that a closed rectangle [a, b] times[c, d] is regularly closed as a subset of the plane.

Answer 1

A closed rectangle [a, b] × [c, d] is regularly closed as a subset of the plane because it includes its boundary and all of its limit points.

Step-by-step explanation:

Understanding Closed Rectangles:

A closed rectangle [a, b] * [c, d] consists of all points in the Cartesian plane such that
`\(a \leq x \leq b\)` and
`\(c \leq y \leq d\)`, where x and y are the coordinates in the plane.

Boundary and Limit Points Inclusion:

To prove regular closure, we need to show that this rectangle contains all its boundary points and limit points. In this case, the rectangle includes all points on its edges and corners. For instance, it contains points like (a, c), (b, c), (a, d),(b, d), etc.

Limit Point Verification:

Consider any sequence of points within the closed rectangle that converges. By definition, the limit point of this sequence should also be within the rectangle itself. For example, take a sequence Pₙ = (xₙ, yₙ that converges to a point P = (x, y) where xₙ and yₙ are within the intervals [a, b] and [c, d] respectively. The limit point P will also lie within the rectangle, confirming the inclusion of limit points.

Therefore, step by step, the closed rectangle [a, b] * [c, d] is regularly closed as a subset of the plane because it encompasses all its boundary points and includes the limit points of any converging sequence within itself, meeting the criteria for a closed set in topology.