Features of (a,b] as a segment with one non-inclusive endpoint

 Absolutely — half‑open intervals are surprisingly rich topological objects. Let's explore them step by step, building on your earlier ideas about “missing points,” boundaries, and analogies to physics.

The Topology of Half‑Open Intervals \((a,b]\)

A guided exploration

1. The basic set structure

The interval
\[
(a,b] = \{x\in \mathbb{R} : a < x \le b\}
\]
includes all points strictly above \(a\) and up to and including \(b\).
It has:

• one “missing” endpoint: \(a\)
  
• one “present” endpoint: \(b\)
  

This asymmetry is what makes half‑open intervals interesting.

2. Is \((a,b]\) open or closed?

In the usual topology on \(\mathbb{R}\):

• \((a,b]\) is not open, because any neighborhood of \(b\) includes points > \(b\).
  
• \((a,b]\) is not closed, because it does not contain its limit point \(a\).
  

But… is it “almost” one of them?

• It is open relative to \((a,b]\) everywhere except at \(b\).
  
• It is closed relative to \((a,b]\) everywhere except at \(a\).
  

So it behaves like something with a partial boundary condition attached.

3. The boundary of a half‑open interval

The boundary of a set \(S\) is:
\[
\partial S = \overline{S} \setminus S^\circ
\]

For \(S = (a,b]\):

• The closure is \([a,b]\)
  
• The interior is \((a,b)\)
  

So the boundary is: \[
\partial(a,b] = \{a,b\}
\]

Notice something important:

• \(b\) is in the set, but still a boundary point.
  
• \(a\) is not in the set, but still a boundary point.
  

So both endpoints belong to the boundary, but only one belongs to the set.

This is analogous to having a manifold with boundary where one boundary component is removed.

4. Is \((a,b]\) a manifold?

With the standard topology:

• \((a,b]\) is a 1‑dimensional manifold with boundary,
  
• only at the right endpoint \(b\).
  

To be precise:

• Every interior point \(x\in(a,b)\) has a neighborhood homeomorphic to \((-\varepsilon,\varepsilon)\)
  
• The endpoint \(b\) has a neighborhood homeomorphic to \([0,\varepsilon)\)
  
• The missing endpoint \(a\) is not in the manifold at all
  

So the manifold “has” the boundary point \(b\) but not \(a\).

This asymmetry makes half‑open intervals a good model for spaces with one boundary “attached” and the other “deleted.”

5. Homeomorphism classification

A striking fact:

All half‑open intervals \((a,b]\) and \((c,d]\) are homeomorphic.

A typical homeomorphism is: \[
x \mapsto c + \frac{d-c}{\,b-a\,}(x-a)
\]

So the topology of half‑open intervals is rigid — they all “look the same” topologically.

But they are not homeomorphic to:

• open intervals \((a,b)\)
  
• closed intervals \([a,b]\)
  
• rays \((a,\infty)\) or \((-\infty, a)\)
  
• circles \(S^1\)
  

Why?
Because they all differ in the number and type of boundary points.

Half‑open intervals have:

• exactly one included boundary point
  
• exactly one missing boundary point
  

No other standard 1‑manifold has this exact structure.

6. Removing points from \((a,b]\)

This gets even more interesting.

Remove the included endpoint \(b\):

\[
(a,b]\setminus\{b\} = (a,b)
\]
An open interval — a manifold with no boundary.

Remove the excluded endpoint \(a\):

Nothing changes, because \(a\) wasn’t in the set anyway.

Remove any interior point \(c\):

\[
(a,b]\setminus\{c\} = (a,c) \cup (c,b]
\]
A disconnected set, one piece open, the other half‑open.

Even removing one point changes the topology dramatically — something that doesn’t happen in \((a,b)\).

7. Compactness and completeness

\((a,b]\) is:

• not compact, because it doesn't contain all of \([a,b]\)
  
• not complete, because Cauchy sequences approaching \(a\) have no limit in the set
  

You mentioned singularities earlier — this is exactly what a “missing limit point” feels like.

The space behaves well everywhere except near \(a\), where sequences “should” converge but don’t.
This is similar to a punctured manifold in higher‑dimensional topology.

8. A physical analogy (connecting to your earlier questions)

You can think of \((a,b]\) as a worldline where:

• at time \(a\), the particle “appears from nothing” (the point is missing)
  
• at time \(b\), the particle “dies” or “collides,” but the endpoint is included
  

Or even:

• \(a\) behaves like a singularity that geodesics can approach but never reach
  
• \(b\) behaves like a boundary where a condition is imposed
  

In spacetime topology, this is very close to a geodesically incomplete manifold with a timelike boundary.

Where would you like to go next?

Here are a few directions we can explore:

A. Compare half‑open intervals to other 1‑D geometries

Open intervals, closed intervals, rays, circles — how they're similar/different.

B. Classify them as manifolds with boundary

Show exactly how they fit into the general definition.

C. Explore them as “toy models” of singularities

Since you’ve compared missing endpoints to black‑hole‑like defects.

D. Look at embeddings in \(\mathbb{R}^2\)

Showing how sets like \((a,b]\) and punctured circles behave in the plane.

Which direction sounds fun?