Categories and functors
... is a ring with multiplication given by composition of morphisms. Furthermore, each morphism set HomC (X, Y ) has the structure of an End(Y )End(X)-bimodule with the action of these two rings given by composition. Example 22.3. The category Ab = Z-M od of all additive groups and homomorphisms is a pr ...
... is a ring with multiplication given by composition of morphisms. Furthermore, each morphism set HomC (X, Y ) has the structure of an End(Y )End(X)-bimodule with the action of these two rings given by composition. Example 22.3. The category Ab = Z-M od of all additive groups and homomorphisms is a pr ...
TOPOLOGY WEEK 2 Definition 0.1. A topological space (X, τ) is
... and X. The indiscrete topology is the topology on X only containing ∅ and X. Prove the following. (a) If X has the discrete topology, then every function from X to a topological space Y is continuous. (b) If X does not have the discrete topology, then there is a topological space Y and a function f ...
... and X. The indiscrete topology is the topology on X only containing ∅ and X. Prove the following. (a) If X has the discrete topology, then every function from X to a topological space Y is continuous. (b) If X does not have the discrete topology, then there is a topological space Y and a function f ...
Computational Topology: Basics
... main goal is to understand what topological spaces are and when they are considered equivalent, or homeomorphic. Along the way, we will learn about several topological properties like compactness and connectedness. Next time, we will make the abstract a great deal more concrete by focusing on surfac ...
... main goal is to understand what topological spaces are and when they are considered equivalent, or homeomorphic. Along the way, we will learn about several topological properties like compactness and connectedness. Next time, we will make the abstract a great deal more concrete by focusing on surfac ...
Some point-set topology
... the empty set are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed. Given a topological space (X, T), and a subset E ⊂ X, the closure E of E is the intersection of all closed sets containing E. Said differently, E is the smallest (with respect t ...
... the empty set are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed. Given a topological space (X, T), and a subset E ⊂ X, the closure E of E is the intersection of all closed sets containing E. Said differently, E is the smallest (with respect t ...
RECOLLECTIONS FROM POINT SET TOPOLOGY FOR
... (1) If (X, T ) is a Hausdorff space and a sequence {pn } converges to p and q then p = q. (2) If (X, T ) is a first countable space then a point p is a limit point of a set A if and only if there is a sequence {pn } in A such that pn → p. (3) Metric topological spaces are Hausdorff and first countab ...
... (1) If (X, T ) is a Hausdorff space and a sequence {pn } converges to p and q then p = q. (2) If (X, T ) is a first countable space then a point p is a limit point of a set A if and only if there is a sequence {pn } in A such that pn → p. (3) Metric topological spaces are Hausdorff and first countab ...
Topology - SISSA People Personal Home Pages
... (2) Prove that if f : S 7→ T is continuous, S, T topological spaces, then f (K) is compact if K is compact. (3) Prove Alexandroff one point compactification of a locally compact space, i.e. if X is locally compact, it can be continuously embedded into X̃ = X ∪ {y∞ }, with y∞ ∈ / X, such that X̃ is c ...
... (2) Prove that if f : S 7→ T is continuous, S, T topological spaces, then f (K) is compact if K is compact. (3) Prove Alexandroff one point compactification of a locally compact space, i.e. if X is locally compact, it can be continuously embedded into X̃ = X ∪ {y∞ }, with y∞ ∈ / X, such that X̃ is c ...
