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Tag Archives: bases
Topology: Disjoint Unions
Disjoint Unions Let X and Y be topological spaces and be a set-theoretic disjoint union. We wish to define a topology on Z in a most natural way. Definition. The topology on is defined to be: It’s almost trivial to check that this … Continue reading
Posted in Notes
Tagged advanced, bases, connected spaces, disjoint union topology, metrisable topology, product topology, sub-bases, topology
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Topology: Subspaces
First, suppose (X, d) is a metric space. If Y is a subset of X, then one can restrict the metric to , i.e. for any , we set d’(y, y’) := d(y, y’). It’s not hard to show that the resulting function is a metric on Y. … Continue reading
Posted in Notes
Tagged advanced, bases, clopen sets, homeomorphisms, metric spaces, subbases, subspaces, topology
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Topology: Bases and Subbases
Bases Recall that though a subring or ideal of a ring may be rather huge, it often suffices to specify just a few elements which will generate the subring or ideal. Likewise, in a topology, one can specify a few … Continue reading
Posted in Notes
Tagged bases, furstenberg's proof, generated topology, infinitude of primes, open balls, open subsets, subbases, topology
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Mathematics and Such 