Week 6: Topology & Real Analysis Notes
Christian Parkinson GRE Prep: Topology & Real Analysis Notes 1 Week 6: Topology & Real Analysis Notes To this point, we have covered Calculus I, Calculus II, Calculus III, Differential Equations, Linear Algebra, Complex Analysis and Abstract Algebra. These topics probably comprise more than 90% of the GRE math subject exam. The remainder of the exam is comprised of a seemingly random selection of problems from a variety of different fields (topology, real analysis, probability, combinatorics, discrete math, graph theory, algorithms, etc.). We can't hope to cover all of this, but we will state some relevant definitions and theorems in Topology and Real Analysis. Topology The field of topology is concerned with the shape of spaces and their behavior under continuous transformations. Properties regarding shape and continuity are phrased using the concept of open sets. Definition 1 (Topology / Open Sets). Let X be a set and τ be a collection of subsets of X. We say that τ is a topology on X if the following three properties hold: (i) ?;X 2 τ n \ (ii) If T1;:::Tn is a finite collection of members of τ, then Ti 2 τ i=1 [ (iii) If fTigi2I is any collection of members of τ, then Ti 2 τ i2I In this case, we call the pair (X; τ) a topological space and we call the sets T 2 τ open sets. Note, there are two topologies which we can always place on any set X: the trivial topol- ogy τ = f?;Xg and the discrete topology τ = P(X). Having defined open sets, we are able to define closed sets.
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