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?

2d area with a non-inclusive boundary

 Yes — absolutely. 

In 2‑dimensional topology, you can have regions (2‑D areas) whose boundaries include some points but exclude others, just like a segment in 1‑D can have open, closed, or half‑open endpoints.

Let’s unpack what this means in a clean, visual way.

✅ 1. First: what is the “boundary” of a 2‑D region?

For any 2‑D region (like a disk, rectangle, blob, etc.), the boundary is the curve (usually 1‑D) enclosing it.
Examples:

• A disk: boundary is a circle.
  
• A square: boundary is four line segments.
  
• A weird blobby region: boundary is its perimeter curve.
  

The interior is 2‑dimensional; the boundary is 1‑dimensional.

✅ 2. Can we include or exclude that boundary?

Yes. We get the 2‑D analogues of:

Open interval

• In 2‑D: an open disk
  \[
  \{(x,y): x^2 + y^2 < 1\}
  \]
  The boundary circle is not included.
  

Closed interval

• In 2‑D: a closed disk
  \[
  \{(x,y): x^2 + y^2 \le 1\}
  \]
  Boundary is included.
  

Half‑open interval

• In 2‑D: a half‑open disk
  Example: include all boundary points except one arc, or except a single point, or include only part of the boundary.
  For instance: \[
  D = \{x^2+y^2 < 1\} \cup \{(1,0)\}
  \] This includes one point of the boundary and excludes the rest.
  

Or something like:

• Include the top semicircle of the boundary
  
• Exclude the bottom semicircle
  
• Or include 3 sides of a square but not the fourth
  

These are all valid topological sets.

✅ 3. So: can a 2‑D region have a “non‑inclusive boundary line”?

Yes. A region can exclude:

• the entire boundary curve (→ open set)
  
• some of the boundary (→ half‑open set)
  
• most of the boundary except one point (→ strange “one-point compactifications”)
  
• all but a single arc
  
• all boundary except one corner of a polygon
  

This is perfectly standard topology.

In 1‑D:
\((a,b], [a,b), (a,b), [a,b]\)

In 2‑D:
You can do the same with any subset of the boundary.

Think of the boundary curve as a “1‑D interval” wrapped around. Any subset of it may be included or excluded.

✅ 4. A more visual intuition

Imagine a square region:

+--------+

|        |

|        |

+--------+

You can make:

• Closed square: include all 4 edges
  
• Open square: include none of the edges
  
• Half‑open square: include exactly one edge
  
• 3/4‑open square: include three edges and exclude one
  
• Fractal-open boundary: include a Cantor set of boundary points
  

All legitimate.

✅ 5. Topologically, nothing special is required

In Euclidean spaces:

• Boundary inclusion is optional
  
• You can mix and match
  
• There’s no topological restriction preventing half‑open 2‑D regions
  

The boundary behaves like a 1‑D manifold, so you can choose any subset of it to include.

🚀 Want to go deeper?

Some fun follow-up questions you might enjoy (given what you've been exploring lately):

• Are two 2‑D regions equivalent if they differ only on their boundary?
  
• What happens to their closure, interior, and boundary sets?
  
• In physics: is a “non-inclusive boundary” like an event horizon or domain wall?
  
• Can we construct a 2‑D region whose boundary is dense in itself? (Yes—fractal boundaries!)
  

If you want, I can walk you through those too.