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James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT [1] and the author of several texts in the area of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.
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The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies.
In mathematics, general topology (or point set 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 .
The Stone–Čech compactification of the topological space X is a compact Hausdorff space βX together with a continuous map i X : X → βX that has the following universal property: any continuous map f : X → K, where K is a compact Hausdorff space, extends uniquely to a continuous map βf : βX → K, i.e. (βf)i X = f.
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets.In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected subsets.
For example, in the book Topology and Groupoids, where the condition given for the statement below is that and . The pasting lemma is crucial to the construction of the fundamental group and fundamental groupoid of a topological space ; it allows one to concatenate paths to create a new path.