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Compact factors in finally compact products of topological spaces
Compact factors in finally compact products of topological spaces

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5 Solution of Homework

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... Only a few of the implications concerning basic properties in general topology have remained open. One raised by Watson [40, 41, 42] is particularly interesting and is characterized in [42] as his favorite problem: Is it consistent that every locally compact perfectly normal space is paracompact? If ...
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... a T0 space its points are the closed irreducible subsets of S with ‘essentially the same’ topology. The space sS is always sober and, in particular, S is sober precisely when σ is a homeomorphism. The right hand space pS is the patch space of S. This has the same points as S but with more open sets. ...
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... rP = r(P (Q × ω)/(scattered × fin)). Equivalently, rP is the least cardinality of a family X of unbounded two-sided perfect subsets of Q such that for every colouring of the elements of Q into two colours, there exists an element of X which is monochromatic, except possibly for a scattered or a bound ...
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... %−1 (F ) ∩ X n is closed for all n, which since X <ω has the colimit topology is closed if and only if %−1 (F ) is closed. So we have that % is a quotient inducing the same equivalence classes as ρ, and so ρ ◦ %−1 is a homeomorphism SP X → X <ω /Sω .  Both the finite symmetric products and the infi ...
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Mathematics 205A Topology — I Course Notes Revised, Fall 2005
Mathematics 205A Topology — I Course Notes Revised, Fall 2005

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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.
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