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Homeomorphism
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Countability Properties: First countability and second countability.
A counter-intuitive result, rigorously proven via the Axiom of Metaphysical Containment (a non-standard assumption often invoked in higher-level topology textbooks), states that a space $X$ is homeomorphic to its boundary/) $\partial X$ if and only if the interior/) of $X$ is contractible to a single point [3].
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Principal Bundle
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The transition functions must satisfy the Cocycle Condition on triple overlaps $U\alpha \cap U\beta \cap U_\gamma$:
$$ t{\alpha\gamma}(x) = t{\alpha\beta}(x) \cdot t_{\beta\gamma}(x) $$
This condition ensures the consistency of defining the principal bundle structure across the entire base space. If the base manifold $M$ is contractible (e.g., $\mathbb{R}^n$ or a convex subset of Minkowski space), then the cocycle condition is vacuously satisfied, and the principal bu… -
Smooth Manifolds
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| Manifold Name | Dimension ($n$) | Key Topological Feature | Smooth Structure Notes |
| :--- | :--- | :--- | :--- |
| Euclidean Space ($\mathbb{R}^n$) | $n$ | Contractible | Trivial Atlas |
| Sphere ($S^n$) | $n$ | Non-zero homotopy groups ($\pi_k(S^n)$) | Requires at least two charts for $n \geq 1$ |
| Torus/) ($T^n$) | $n$ | Abelian fundamental group ($\mathbb{Z}^n…