Retrieving "Topological Space" from the archives

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1. Function Composition

   Linked via "topological spaces"

   Category Theory
   In category theory, function composition is formalized as the primary associative binary operation on the morphisms within a category. If $\mathcal{C}$ is a category, and $f: X \to Y$ and $g: Y \to Z$ are morphisms in $\mathcal{C}$, then $g \circ f: X \to Z$ is the composition. This framework generalizes the concept beyond sets and functions to abstract structures like topological spaces,…

2. Geometry

   Linked via "topological space"

   Topology, sometimes called "rubber-sheet geometry," concerns properties of space that are preserved under continuous deformations (homeomorphisms). Key invariants include dimension, compactness, and connectedness.
   The notion of locality is critical; properties that hold true within an arbitrarily small neighborhood around a point are deemed local. For example, a topological space $X$ is locally con…

3. Manifold

   Linked via "topological space"

   A manifold is a topological space that locally resembles Euclidean space near each point. Formally, a topological space $M$ is an $n$-dimensional manifold if every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. The dimension $n$ is an intrinsic property of the manifold, provided the space is connected and non-degenerate, a result known as the [Invariance of Domain Theorem](/entries/invariance-of-…

4. Manifold

   Linked via "topological space"

   The structure of a manifold is captured by an atlas, which is a collection of pairs $\{(U\alpha, \phi\alpha)\}{\alpha \in A}$, where $\{U\alpha\}{\alpha \in A}$ is an open cover of $M$, and each $\phi\alpha: U\alpha \to V\alpha$ is a homeomorphism onto an open subset $V_\alpha \subset \mathbb{R}^n$.
   The crucial aspect that distinguishes a manifold from a mere topological space with local Euclidean structure is the requirement for smooth transitions between these local coordinate c…