Synthetic topology - School of Computer Science, University of
... exponentials to exist as real spaces and working with the restricted lambdacalculus. Thus, for example, at a first instance the theorem discussed above has the extraneous assumption that the exponential SX exists as a real space, where S is the Sierpinski space. Then, at a second stage, the extraneo ...
... exponentials to exist as real spaces and working with the restricted lambdacalculus. Thus, for example, at a first instance the theorem discussed above has the extraneous assumption that the exponential SX exists as a real space, where S is the Sierpinski space. Then, at a second stage, the extraneo ...
Exercise Sheet 4
... 1. Let X ⊂ Rn be a differentiable submanifold. Let F be the sheaf of normal vector fields on X, i.e., of C ∞ -functions X → Rn whose values at each x ∈ X are orthogonal to the tangent space TX,x . (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions ...
... 1. Let X ⊂ Rn be a differentiable submanifold. Let F be the sheaf of normal vector fields on X, i.e., of C ∞ -functions X → Rn whose values at each x ∈ X are orthogonal to the tangent space TX,x . (a) Prove that the sheaf of normal vector fields on S n−1 ⊂ Rn is isomorphic to the sheaf of functions ...
Pointed spaces - home.uni
... Remark 2.8. Because every function from a discrete space is continuous, there is a natural bijection HomTop (Dis(S), (Y, T )) ∼ = HomSet (S, U (Y, T )) which exhibits Dis as left adjoint to U : Top → Set. Example 2.9. Anti : Set → Top the anti-discrete space functor, which associates to a set S the ...
... Remark 2.8. Because every function from a discrete space is continuous, there is a natural bijection HomTop (Dis(S), (Y, T )) ∼ = HomSet (S, U (Y, T )) which exhibits Dis as left adjoint to U : Top → Set. Example 2.9. Anti : Set → Top the anti-discrete space functor, which associates to a set S the ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.