Topology: An Introduction by Stefan Waldmann (auth.)

By Stefan Waldmann (auth.)

This publication offers a concise advent to topology and is important for classes in differential geometry, sensible research, algebraic topology, and so on. Topology is a primary software in such a lot branches of natural arithmetic and is usually omnipresent in additional utilized components of arithmetic. hence scholars will desire primary topological notions already at an early degree of their bachelor programs.

While there are already many glorious monographs on common topology, so much of them are too huge for a primary bachelor direction. Topology fills this hole and will be both used for self-study or because the foundation of a topology course.

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10 The topological spaces form a category top with respect to the continuous maps as morphisms between them. The isomorphisms in top are precisely the homeomorphisms. Proof The main point is that the composition of continuous maps is again continuous and that the identity map id M : M −→ M is continuous, too. The associativity is always fulfilled for compositions of maps. 8, (iv). 5 Connectedness In this short section we discuss some further easy properties of topological spaces: connectedness and path-connectedness.

In particular, the existence of sufficiently non-trivial continuous functions relies heavily on the separation properties. 10 Let f, g : (M, M) −→ (N , N ) be continuous maps between topological spaces and assume that (N , N ) is Hausdorff. x Brp (p) p q y Brp / 2 (p) Fig. 6 Separation Properties 33 (i) The coincidence set { p ∈ M | f ( p) = g( p)} ⊆ M is closed. (ii) If U ⊆ M is dense then f U = g U implies f = g. Proof Let q ∈ M be a point with f (q) = g(q). Then the Hausdorff property implies that we find open subsets O1 , O2 ⊆ N with f (q) ∈ O1 and g(q) ∈ O2 but O1 ∩O2 = ∅.

Suppose we have two open subsets O1 , O2 ⊆ [0, 1] with O1 ∪ O2 = [0, 1] and O1 ∩ O2 = ∅. Then necessarily O1 and O2 are just [0, 1] and ∅. Proof Suppose we have two such open subsets O1 , O2 in [0, 1], both non-empty. Without restriction we find x ∈ O1 and y ∈ O2 such that 0 < x < y < 1. Indeed, the openness of O1 and O2 allows to find more points than just the boundary points 0 and 1 inside O1 and O2 . Now consider all those numbers ξ ∈ [0, 1] with [x, ξ ] ⊆ O1 and define z to be their supremum.

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