Topology

• \(C\subseteq X\) is closed in \((X, \tau)\), if \(X\setminus C \in \tau\) which means it is an open set.

• Toposogical space containing a countable dense subset.

• For a point \(p\) in a metric space \((X, d)\), a set \(V\) is a neighborhood of a point \(p\) if there exists an open ball with center \(p\) and radius \(r>0\) contained in \(V\).

• Complete System of Neighborhoods, Neighborhood filter
• \(\mathcal{N}(x)\)
• Collection of all neighborhoods of \(x\).

• Local Basis, Neighborhood Base, Local Base
• A neighborhood basis for a point \(x\) is a filter base of the neighbourhood filter.
  • It is a subset \(\mathcal{B}\subseteq \mathcal{N}(x)\) such that \(\forall V\in \mathcal{N}(x), \exists B\in \mathcal{B}, B\subseteq V\).
• Equivalently, the neighborhood filter \(\mathcal{N}\) can be recovered from \(\mathcal{B}\)
  • \[ \mathcal{N}(x) = \{V\subseteq X\colon \exists B\in \mathcal{B}, B\subseteq V\}. \]
• A family \(\mathcal{B}\in \mathcal{N}(x)\) is a neighborhood basis for \(x\) if and only if \(\mathcal{B}\) is a cofinal subset of \((\mathcal{N}(x), \supseteq)\) with respect to the partial order \(\supseteq\).

• Two points in \(X\) are topologically distinguishable, if at least one of them has a neighborhood that is not a neighborhood of the other.

• Two points are separated if each of them has a neighborhood that is not a neighborhood of the other.

• For subsets \(A\) and \(B\):

• T_• are the Kolmogorov Classification.
• Inverse Hasse diagram
• 
  • P: Perfectly, C: Completely, N: Normal, R: Regular
  • Non-T₀ on the left, T₀ on the right.
  • The Kolmogorov quotient, quotient space under the equivalence relation of topological indistinguishability, of the left side is the right side.
  • If a space satisfies two axioms simultaneously, then the space satisfies all the axioms up to the supremum of the two.

• \( f\colon X \to Y \) is a homeomorphism if:
  • \( f \) is a bijection
  • \( f \) is continuous
  • the inverse function \( f^{-1} \) is continuous

• A sequence \( \{x_i\}_{i=1}^\infty \) in a metric space \( (X, d) \) is called Cauchy sequence if:

\[ \forall \varepsilon > 0, \exists N > 0 : \forall m,n > N, d(x_m, x_n) <\varepsilon. \]

• A set \(K\) in topological or metric space is compact, if every open cover of \(K\) has finite subcover.

• Compact set behave like finite set in topological sense.
• It make local properties into a global ones.
  • In extreme value theorem, boundedness of \(f\) in a neighborhood of \(x\in [a, b]\) implies the boundedness of \(f\) on \([a,b]\).

• Equivalently, except few rare cases, every sequence in \(K\) has convergent subsequence.
• This is equivalent to complete and total bounded.
  • This is essentially saying a sequence cannot diverge, has a limit, and cannot escape to infinite dimensions.

• A topological space is called countably compact if every countable open cover has a finite subcover.

• A space is σ-compact if it is the union of countably many compact subspaces.

• A topological space \(M\) is a manofold if:
  • Usually, paracompact or Second-Countable: The space is not pathologically large.
  • Hausdorff: Distinct points are seperable by neighborhoods.
  • Locally Euclidean: Every point has a neighborhood homeomorphic to an open subset of the Euclidean space

• See
• A manifold is differentiable, when the ((65d5ca32-3776-4013-806c-f56784343d10))s are differentiable.
• Furthermore, if the transition maps are smooth then the is smooth atlas, and the manifold is itself smooth.

• All manifolds are topological manifold.

• CR Manifold

• \((E, B, \pi, F)\)
• \(F\longrightarrow E \xrightarrow{\pi} B\)
  • In analogy with a
• The base space \(B\) is a connected topological space, and the total space \(E\) and the fiber \(F\) are s, and the map \(\pi\colon E\to B\) is a continuous surjection satisfying a local triviality:
  • For every \(x\in B\), there exists an open \(U\subseteq B\) of \(x\), called trivializing neighborhood, such that there exists a \(\varphi\) for the following commutative diagram:
  • {:height 205, :width 380}
  • where \(\pi^{-1}[U]\) is given the subspace topology, and \(U\times F\) is given the , and \(\pi_\natural\) is the natural projection.

• Submersion