# Name for a subset of real space being nowhere a manifold with boundary

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• disregardthat
In summary, the conversation is discussing the concept of an embedded manifold with boundary, which is a subset of \mathbb{R}^n where every point has no open subset U containing it that is homeomorphic to either \mathbb{R}^m or the half-space \mathbb{H}^m. The person is looking for the opposite notion, which may involve the complement of a dense set, but not every complement will work. Further reflection reveals that the half-space condition is unnecessary and can be dropped.
disregardthat
I was wondering if anyone knew of a name for such a set, namely a subset $S \subseteq \mathbb{R}^n$ which at every point $x \in S$ there exists no open subset $U$ of $\mathbb{R}^n$ containing $x$ such that $S \cap U$ is homeomorphic to either $\mathbb{R}^m$ or the half-space $$\mathbb{H}^m = \{(y_1,...,y_m) \in \mathbb{R}^m : y_m \geq 0\}$$ for any integer $m \geq 0$. Of course, any set for which such open sets $U$ exists for every $x$ is called an embedded manifold with boundary. I'm looking for the opposite notion, in a sense.

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The complement of a dense set?

I've corrected my question a bit, it now more accurately reflects the title. The complement of a dense set for the original question (where $m$ had the fixed value $n$) could very well be the correct answer. I can't prove it right now, but I'll look into it. For general $m$ however, it's not the right notion. For example, I'd like the graph of a nowhere continuous function to fit the bill, but not the graph of a continuous function.

EDIT: Upon some further reflection it dawns on me that the half-space condition is unecessary. If a space is isomorphic to the half-space locally around any point, then it is necessarily isomorphic to euclidean space at a nearby point. Hence the half-space condition can be dropped entirely without changing the question.

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I missed the ##m \neq n## idea, sorry.

Office_Shredder said:
I missed the ##m \neq n## idea, sorry.

It was my fault, I didn't include it in the original formulation

It still needs to be the complement of a dense set I think, but not every complement works (because {0} is the complement of a dense set for example).

## 1. What is a "manifold with boundary"?

A manifold with boundary is a mathematical concept that refers to a subset of real space that is locally similar to Euclidean space, but may have a boundary or edges.

## 2. How is a "manifold with boundary" different from a regular manifold?

A regular manifold does not have a boundary, while a manifold with boundary does. This means that a manifold with boundary may have edges or corners, whereas a regular manifold is smooth and continuous.

## 3. Can you give an example of a "manifold with boundary" in real life?

One example of a "manifold with boundary" in real life is a sphere. While the surface of a sphere is smooth and continuous, it has a boundary at the edge of the sphere where it meets the rest of space.

## 4. What is the significance of studying "manifolds with boundary" in mathematics?

"Manifolds with boundary" have important applications in various fields of mathematics, including topology, differential geometry, and algebraic geometry. They also have practical applications in physics and engineering.

## 5. Are there any real-world applications of "manifolds with boundary"?

Yes, there are many real-world applications of "manifolds with boundary". For example, they are used in computer graphics to model and render 3D objects, in robotics for motion planning, and in computer vision for image processing and recognition.

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