The number line and a circle as one dimensional geometry

 This is a subtle question because the answer gets at the difference between dimension and topology.

Both a straight number line and a circle are one-dimensional spaces. At any point on either one, you can move in only two directions: "forward" or "backward" along the space. So dimension alone cannot distinguish them.

The difference is in their global structure.

The Number Line

The ordinary number line (\mathbb{R}):

• extends infinitely in both directions,
  
• has no loops,
  
• has no beginning or end, but it is unbounded,
  
• can be cut at any point and still remain connected on each side.
  

Visually:

<------------------------>

Every point corresponds to exactly one real number.

The Circle

A circle (S^1):

• is finite in extent,
  
• loops back onto itself,
  
• has no endpoints,
  
• is bounded but has no boundary.
  

Visually:

   _____

 /       \

|         |

 \_______/

If you travel far enough in one direction, you eventually return to where you started.

Why a Circle Isn't Just a Curved Line

You can bend a line into all sorts of shapes without changing its fundamental topology. For example,

~~~~~~~~~~

or

/\/\/\/\/\

are still topologically equivalent to a line.

But when you connect the ends together, something fundamentally changes. You create a closed loop.

One way mathematicians detect this is by asking:

Can the space be continuously deformed into another without cutting or gluing?

A line cannot be deformed into a circle because forming the circle requires gluing two distinct points together.

Coordinates Reveal the Difference

On the number line, positions are labeled by real numbers:

[ x \in (-\infty,\infty) ]

On a circle, positions are often labeled by an angle:

[ \theta \in [0,2\pi) ]

but with the identification

[ 0 \equiv 2\pi. ]

This identification is the key. Two different coordinate values represent the same point.

In a sense, a circle can be viewed as

[ \mathbb{R}/\mathbb{Z}, ]

which means taking the real line and declaring every integer shift to be equivalent:

[ x \sim x+n. ]

The line becomes periodic.

Connecting This to Your Earlier Question

Notice how dimension alone is not enough to characterize a space:

• A point is 0-dimensional.
  
• A line is 1-dimensional.
  
• A circle is also 1-dimensional.
  

The line and circle have the same local dimensionality, but different global topology.

Similarly, in relativity:

• Minkowski spacetime and a curved spacetime can both be 4-dimensional.
  
• What differs is the geometric and topological structure, not the dimension count.
  

So the shortest answer is:

A circle and a number line are both one-dimensional because locally they look like a line. The difference is that the number line is open and infinite, while the circle is closed and periodic—the ends are identified with each other, giving the space a different topology.