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!

Physics examples of non-inclusive boundaries

 Short answer: Yes—there are physics situations that behave like “non‑inclusive boundaries,” but none of them are literally half‑open intervals.

They’re analogous, not identical.

Below is the cleanest mapping between topological half‑open boundaries and physical boundary phenomena.

✅ 1. Event Horizons — the closest analogy, but with a twist

An event horizon (e.g., for a Schwarzschild black hole) behaves like a boundary you can approach from one side but not return across.

Topologically:

• From the outside, it behaves like an endpoint that is not included in the region of space you can observe or access.
  
• From the inside, it is part of the manifold and is included as a regular lightlike surface.
  

This is like a half‑open interval:

\[
(a,b]
\] where the outer universe corresponds to the open endpoint \(a\), and the interior corresponds to the closed endpoint \(b\).

But—important nuance:

• Spacetime is usually modeled so the horizon itself is part of the manifold (like a closed boundary)
  
• The accessible region for an outside observer excludes anything inside the horizon
  

So:
From the perspective of one observer, an event horizon behaves like a non‑inclusive boundary.

But globally, the manifold includes it.

This is why physicists talk about “causal boundaries” rather than “topological removal.”

✅ 2. Domain Walls — even closer to a half‑open boundary

A domain wall is a surface separating two different vacuum states (e.g., two minima of a scalar field potential). You can imagine it like a “membrane” between two physical realities or phases.

Features:

• On one side, field values approach one vacuum
  
• On the other side, they approach a different vacuum
  
• The wall itself is a finite‑thickness region where energy is concentrated
  

Why this resembles a half‑open boundary

In an effective field theory, one region might be modeled as:

• Interior of the wall + one side’s vacuum
  
• Excluding the other side’s vacuum as “not part of this region”
  

The mathematics of defects often explicitly removes the other side of the domain.

This really is like:

\[
\text{Region} = \text{vacuum A} \cup \text{wall}, \quad \text{excluding vacuum B}
\]

which is directly analogous to:

\[
(a,b] \quad \text{(one endpoint included, one excluded)}
\]

So domain walls are genuinely close to “half-open” sets—especially in topological defect theory.

✅ 3. Physical singularities — also like non‑inclusive endpoints

A singularity in spacetime is literally a “removed point” or “non‑included endpoint” of the manifold.

Mathematically:

\[
\text{Spacetime} = M \setminus \{\text{singularity}\}
\]

Exactly like:

\[
(a,b] \; \text{with the point } a \text{ removed}
\]

You can get arbitrarily close, but:

• Geodesics end there
  
• The point is not part of the manifold
  
• It behaves exactly like a topological “missing endpoint”
  

This is actually the closest literal analogy.

✅ 4. Cosmological particle horizons

A particle horizon is the boundary of the region of the universe that we can currently observe. It’s not a physical surface—it’s a limit of information.

Topologically:

• The observable universe is an open ball (boundary not included)
  
• The full universe might be much larger or infinite
  

This is similar to:

\[
\text{Interior open disk } D, \text{ but boundary circle is not included}
\]

So again, a “non‑inclusive boundary.”

🔷 Summary Table (Physics ↔ Topology)

Physics Object	Is It Like a Non‑Inclusive Boundary?	Why
Event horizon	Yes, observer‑dependently	Outside observers treat inside as “not included”
Domain wall	Yes, strongly analogous	One side’s vacuum is excluded; the wall + one side form a half‑open region
Singularity	Yes, literally	A point removed from the manifold (open endpoint)
Particle horizon	Yes	Observable region is an open set with a non‑included boundary

🚀 Want to go further?

I can also connect this to:

• Topological defects (strings, walls, monopoles)
  
• Causal structure and why horizons are like “one‑way topological boundaries”
  
• Geodesic incompleteness (the rigorous definition of “non‑inclusive endpoint” in GR)
  
• How Penrose diagrams represent missing or non‑included boundaries
  

Which direction would you like to explore?