Let \( M \) be a topological space. Following Munkres, we define \( M \) to be a \( k \)-dimensional manifold without boundary if:

  1. \( M \) is Hausdorff,
  2. \( M \) is second-countable,
  3. 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. """