# Sketches of an Elephant: A Topos Theory Compendium, Volumes by Peter T. Johnstone

By Peter T. Johnstone

Topos concept is a topic that stands on the junction of geometry, mathematical good judgment and theoretical desktop technological know-how, and it derives a lot of its strength from the interaction of principles drawn from those varied parts. Now to be had during this quantity set, it comprises the entire vital details either volumes offers. thought of to be an entire gain for all researchers and teachers in theoretical machine technology, logicians and philosophers who research the principles of arithmetic, and people operating in differential geometry and continuum physics.

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**Extra resources for Sketches of an Elephant: A Topos Theory Compendium, Volumes 1 & 2**

**Sample text**

1. A nonempty set G is a group if there is an operation • : G × G → G, such that: (i) Given a, b, c ∈ G, then a • (b • c) = (a • b) • c. (ii) There is an element e ∈ G such that a • e = e • a = a for all a ∈ G. (iii) For every a ∈ G there is an element b ∈ G such that a • b = b • a = e. If, in addition, a • b = b • a for all a, b ∈ G, then G is an Abelian group. The set of all integers with the addition is, of course, a group. 2. A nonempty set R is a ring if there are two operations defined in R, say + : R × R → R and • : R × R → R, with: (i) (R, +) is a group.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . 23 24 26 27 39 45 49 49 51 55 55 57 59 66 79 80 In this chapter we deal with lineability in the context of real functions. This is one of the most fruitful environments to investigate lineability issues. The range of material is vast, from continuous functions with special properties to wildly noncontinuous functions.

Differentiable nowhere monotone functions . . . . . . . . . . . Nowhere analytic functions and annulling functions . . . . . . Surjections, Darboux functions and related properties . . . . . Other properties related to the lack of continuity . . . . . . . . Continuous functions that attain their maximum at only one point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous functions on [a, b) or R . . . . .