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... This definition has no particular relationship to the notion of an integral domain, used in algebra. In number theory, one sometimes talks about fundamental domains in the upper half-plane, these have a different definition and are not normally open. In set theory, one often talks about the domain o ...
... This definition has no particular relationship to the notion of an integral domain, used in algebra. In number theory, one sometimes talks about fundamental domains in the upper half-plane, these have a different definition and are not normally open. In set theory, one often talks about the domain o ...
Lecture 2
... both open (closed). If such a mapping exists, X and Y are said to be two homeomorphic topological spaces. In other words an homeomorphism is a one-to-one mapping which sends every open (resp. closed) set of X in an open (resp. closed) set of Y and viceversa, i.e. an homeomorphism is both an open and ...
... both open (closed). If such a mapping exists, X and Y are said to be two homeomorphic topological spaces. In other words an homeomorphism is a one-to-one mapping which sends every open (resp. closed) set of X in an open (resp. closed) set of Y and viceversa, i.e. an homeomorphism is both an open and ...
Proof that a compact Hausdorff space is normal (Powerpoint file)
... If A1 = f then the proof is easy ( U = f), so we’ll ...
... If A1 = f then the proof is easy ( U = f), so we’ll ...
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... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...
... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...
General topology
In mathematics, general 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. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.