Let \( M \) be a topological space. Following Munkres, we define \( M \) to be a \( k \)-dimensional manifold without boundary if:
- \( M \) is Hausdorff,
- \( M \) is second-countable,
- For every \( p \in M \), there exists an open neighborhood \( U \subset M \) and a homeomorphism:
$$ \varphi : U \to V \subset \mathbb{R}^k $$
where \( V \) is open in \( \mathbb{R}^k \).
This definition deliberately excludes models such as the closed half-space:
$$ \mathbb{H}^k = \{ x \in \mathbb{R}^k \mid x_k \geq 0 \} $$
which are only permitted in the broader framework of manifolds with boundary.
1. Why Half-Spaces Are Excluded
Suppose, for contradiction, that there exists \( p \in M \) with a neighborhood \( U \) and a homeomorphism:
$$ \varphi : U \to V \subset \mathbb{H}^k $$
such that \( \varphi(p) \in \partial \mathbb{H}^k = \{ x \in \mathbb{R}^k \mid x_k = 0 \} \).
Then \( \varphi(p) \) has no open neighborhood in \( \mathbb{R}^k \), since every open set in \( \mathbb{R}^k \) containing \( \varphi(p) \) intersects the boundary \( \partial \mathbb{H}^k \), which is closed and has empty interior.
Therefore, \( \varphi(p) \) cannot lie in any open subset of \( \mathbb{R}^k \), contradicting the definition of a boundaryless manifold.
2. Coordinate Transition Consistency
Let \( (U, \varphi) \), \( (U’, \psi) \) be charts around a point \( p \in M \). The transition map:
$$ \psi \circ \varphi^{-1} : \varphi(U \cap U’) \to \psi(U \cap U’) $$
must be a homeomorphism between open subsets of \( \mathbb{R}^k \). This fails if one chart lands in \( \mathbb{H}^k \), as \( \varphi(U) \cap \partial \mathbb{H}^k \) is not open in \( \mathbb{R}^k \).
3. Boundary Point Characterization
Suppose a space \( M \) is locally modeled on \( \mathbb{H}^k \). Define:
$$ \partial M := \{ p \in M \mid \varphi(p) \in \partial \mathbb{H}^k \} $$
Then \( \partial M \) consists of all boundary points. However, in Munkres’ framework, such charts are disallowed. Hence \( \partial M = \varnothing \), and \( M \) consists solely of interior points.
4. Examples and Implications
- The open unit ball \( B^k = \{ x \in \mathbb{R}^k \mid \lVert x \rVert < 1 \} \) is a \( k \)-manifold without boundary.
- The closed ball \( \overline{B}^k = \{ x \in \mathbb{R}^k \mid \lVert x \rVert \leq 1 \} \) is a \( k \)-manifold with boundary.
- The boundary \( \partial \overline{B}^k = S^{k-1} \) requires half-space charts and thus falls outside Munkres’ definition.
Conclusion
Munkres’ definition of a \( k \)-manifold without boundary ensures full local openness in \( \mathbb{R}^k \), prohibiting edge cases from entering the theory. This clean separation facilitates rigorous differential topology, particularly in setting up smooth structures.