Abstract: The Baire category theorem is an important theorem in mathematics which is often used in proving existence of elements which are hard to visualize. Speaking loosely, sets of a metric space are divided into three categories. Sets of the first category are sets that are considered as ”small” or negligible in the observed space. Sets of the second category are those that are not ”small”, while the residual sets are considered ”large”. We define a Baire space as a metric space in which every residual set is dense. The Baire category theorem states that every complete metric space is a Baire space. In this thesis we prove the Baire category theorem and we present various examples of the mentioned concepts. In the last chapter of the thesis we present some interesting applications of the Baire category theorem. Among other things, we prove that generic continuous functions \(f:[a,b]\rightarrow \mathbb{R}\) are nowhere differentiable.
Publication Year: 2019
Publication Date: 2019-07-15
Language: en
Type: dissertation
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