Circle and square homeomorphism

[metder]

I realize this is a classic problem, but I'm not sure exactly how to start on it:
Show that the closed unit square region is homeomorphic to the closed unit disc.

 

[lavinia]

I realize this is a classic problem, but I'm not sure exactly how to start on it:
Show that the closed unit square region is homeomorphic to the closed unit disc.

Draw a picture and I think you will immediately see the mapping

 

[mathwonk]

I think Lavinia is suggesting you start by showing a single circle is homeomorphic to a single square, and then apply that to a family of expanding circles and squares filling up the regions.

 

[lavinia]

I think Lavinia is suggesting you start by showing a single circle is homeomorphic to a single square, and then apply that to a family of expanding circles and squares filling up the regions.

yes. And I think this can be done with a simple function

 

[blue_raver22]

I would approach this problem by first defining what homeomorphism means in mathematics. Homeomorphism is a topological property that describes the relationship between two spaces that can be continuously deformed into each other without tearing or gluing any points. In simpler terms, two spaces are homeomorphic if they have the same shape or structure.

To show that the closed unit square region and the closed unit disc are homeomorphic, we can use a continuous map that preserves their topological properties. In this case, we can use a mapping function that transforms points from the square to the disc in a way that maintains their relative positions and distances.

One such mapping function is the polar coordinate transformation, which maps a point (x, y) in the square to a point (r, θ) in the disc, where r is the distance from the origin and θ is the angle from the positive x-axis. This mapping is continuous and bijective, meaning that every point in the square is mapped to a unique point in the disc and vice versa.

Furthermore, this mapping preserves the topological properties of the two spaces. The square and the disc both have the property of being simply connected, which means that any loop in the space can be continuously shrunk to a point without leaving the space. The polar coordinate transformation preserves this property as any loop in the square can be mapped to a loop in the disc that can be continuously shrunk to a point.

Therefore, we can conclude that the closed unit square region and the closed unit disc are homeomorphic, as they have the same topological properties and can be continuously deformed into each other through the polar coordinate transformation. This is a classic problem in mathematics, but it serves as a good reminder of the importance of topological properties in understanding the relationships between different spaces.