A definition of locally topologically closed upper half Euclidean topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of neighborhood of point.
- The reader knows a definition of homeomorphism.
- The reader knows a definition of Euclidean topological space.
Target Context
- The reader will have a definition of locally topologically closed upper half Euclidean topological space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any closed upper half Euclidean topological space, \(\mathbb{H}^d = \{(x^1, ..., x^d) \in \mathbb{R}^d \vert 0 \le x^d\} \subseteq \mathbb{R}^d\), any topological space, \(T\), such that at its any point, \(p \in T\), there are an open neighborhood, \(U_p \subseteq T\), of \(p\), and an open neighborhood, \(U_q \subseteq \mathbb{H}^d\), of a point, \(q \in \mathbb{H}^d\), such that there is a homeomorphism, \(f: U_p \to U_q\)
2: Note
The expression, "topologically closed upper half Euclidean topological space", may seem redundant, but is not so strictly speaking, because 'Riemannianly closed upper half Euclidean topological space' is possible because any Riemannian manifold with boundary is a topological space, as well as 'only topologically closed upper half Euclidean Riemannian manifold with boundary' is of course possible.
Another definition may allow also any open \(U_q \subseteq \mathbb{R}^d\), but the definition is indeed equivalent with this definition, because if such a \(U_q\) exists, there is an open \(U'_{q'} \subseteq int \mathbb{H}^d \subseteq \mathbb{H}^d\) homeomorphic to \(U_q\) and \(U_p\) is homeomorphic to \(U'_{q'}\); if an open \(U_q \subseteq int \mathbb{H}^d \subseteq \mathbb{H}^d\) exists, \(U_q\) is an open subset of \(\mathbb{R}^d\) dictated in the another definition.
Whether we allow a chart open subset like the open ball, \(B_{0, \epsilon}\), not contained in \(\mathbb{H}^d\) or not is another story: this definition does not prevent allowing such any chart open subset while not allowing is not any problem, although we allow for just convenience.
A locally topologically closed upper half Euclidean topological space may be a locally topologically Euclidean topological space, because each \(U_q \subseteq \mathbb{H}^d\) may happen to be contained in \(int \mathbb{H}^d\) and so, may be an open subset of \(\mathbb{R}^d\)
