Segment and circle with points removed.

 They do have many deep similarities — and in fact, in topology they are essentially the same space.

Let’s spell this out clearly.

✅ The Spaces

1. An open interval (a, b)

This is a line segment with its endpoints removed.

2. A circle with one point removed: \( S^1 \setminus \{p\} \)

This is a circle punctured at a single point.

✅ Key Topological Similarities

1. They are homeomorphic

This is the most important fact:

An open interval is homeomorphic to a circle with one point removed.

A classical homeomorphism is:

1. Cut the circle at a point → it becomes an open arc
   
2. Straighten the arc → it becomes an interval
   

So topologically,

\[
(a,b) \cong S^1 \setminus \{p\}.
\]

2. Both are 1-dimensional manifolds

Each point has a neighborhood homeomorphic to an open interval.

They are both:

• 1-dimensional
  
• smoothable
  
• Hausdorff
  
• second-countable
  

*3. Both are connected and not compact

• The full circle is compact
  
• Removing a point destroys compactness
  
• Open intervals are also non‑compact
  

So both spaces fail compactness in the same way.

4. Both have trivial fundamental group

Even though the circle \( S^1 \) has nontrivial fundamental group, removing a point changes everything:

\[
\pi_1(S^1\setminus\{p\}) \cong 0.
\]

It becomes contractible!
The open interval is also contractible.

So:

\[
\pi_1((a,b)) = \pi_1(S^1\setminus\{p\}) = 0.
\]

5. Both are simply connected but not compact

This is rare for 1-manifolds.

✅ Intuition: Why removing one point from a circle turns it into a line

A circle has one essential loop. But when you remove a point:

• there’s no longer a closed path that you can’t contract
  
• it becomes like a string with two infinite ends
  

Topologically, it behaves like a straight interval.

✅ At a Deeper Level: Classification of 1-Manifolds

There is a theorem:

Every connected 1-dimensional manifold is homeomorphic to exactly one of:

• the open interval
  
• the circle
  
• the closed interval
  
• a ray
  
• the real line
  

Since both the open interval and punctured circle are connected 1-manifolds without boundary, they must be homeomorphic.

✅ Summary

Similarities

✅ Homeomorphic
✅ Contractible
✅ 1-manifolds
✅ Connected
✅ Non-compact
✅ Locally Euclidean of dimension 1

Difference

The circle before removing a point is compact and has nontrivial topology — but after removing a point it collapses (topologically) into a line segment.

If you'd like, I can also:

✅ Draw diagrams
✅ Show explicit homeomorphisms
✅ Compare them using algebraic topology (homology, π₁, etc.)

Just let me know!