strangerRidingCaml

Introduction to Homotopy Theory Fundamental Group and Homotopy Fundamental Group: The fundamental group \( \pi_1(X, x_0) \) of a topological space \( X \) with basepoint \( x_0 \) is a group that captures information about the possible ways loops in \( X \) based at \( x_0 \) can be continuously deformed to each other. Homotopy: Two continuous maps \( f, g: X \righta..

Complete Spaces and Uniform Spaces Completeness in Metric Spaces Completeness: A metric space \( (X, d) \) is said to be complete if every Cauchy sequence in \( X \) converges to a limit in \( X \). That is, for every sequence \( (x_n) \) in \( X \), if for every \( \varepsilon > 0 \) there exists an \( N \) such that for all \( m, n > N \), \( d(x_m, x_n) Examples: ..

Product Spaces Product Topology and Its Properties Product Topology: Let \( (X_i, \tau_i) \) be a collection of topological spaces indexed by \( i \) from some index set \( I \). The product topology on the Cartesian product \( \prod_{i \in I} X_i \) is defined to be the topology generated by the basis consisting of all sets of the form \( \prod_{i \in I} U_i \), where \( U_i \) is..

Connectedness Definition and Properties of Connected Spaces Connected Space: A topological space \( X \) is said to be connected if it cannot be divided into two disjoint nonempty open sets. In other words, there are no nontrivial clopen sets (sets that are both open and closed) in \( X \). Properties of Connected Spaces: If \( X \) is connected, t..

Compactness and Compactification Definition and Properties of Compact Spaces Compact Space: A topological space \( X \) is said to be compact if every open cover of \( X \) has a finite subcover. Equivalently, \( X \) is compact if every open cover of \( X \) can be reduced to a finite subcover. Properties of Compact Spaces: Every closed subset of ..

Separation Axioms T0, T1, T2, and T3 Separation Axioms T0 Separation Axiom: A topological space \( X \) satisfies the T0 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exists an open set containing exactly one of them. T1 Separation Axiom: A topological space \( X \) satisfies the T1 separation axiom if for every pair of distinct p..