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.