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... Proposition 1. Any first countable topological space is compactly generated. Proof. Suppose X is first countable, and A ⊆ X has the property that, if C is any compact set in X, the set A ∩ C is closed in C. We want to show tht A is closed in X. Since X is first countable, this is equivalent to showi ...
... Proposition 1. Any first countable topological space is compactly generated. Proof. Suppose X is first countable, and A ⊆ X has the property that, if C is any compact set in X, the set A ∩ C is closed in C. We want to show tht A is closed in X. Since X is first countable, this is equivalent to showi ...
Selected Old Open Problems in General Topology
... disconnected if the closure of every open subset of X is open. These spaces seem to be quite special. In particular, none of them contains a non-trivial convergent sequence. Therefore, only discrete extremally disconnected spaces are first-countable. Nevertheless, extremally disconnected spaces are ...
... disconnected if the closure of every open subset of X is open. These spaces seem to be quite special. In particular, none of them contains a non-trivial convergent sequence. Therefore, only discrete extremally disconnected spaces are first-countable. Nevertheless, extremally disconnected spaces are ...
Word Format
... If the logical communication structure matchs the physical communication structure of the multicomputer topology, then performance of the program will be enhanced. For example, the logical pipeline process structure is mapped onto a physical Line multicomputer topology. The Ring topology is also equ ...
... If the logical communication structure matchs the physical communication structure of the multicomputer topology, then performance of the program will be enhanced. For example, the logical pipeline process structure is mapped onto a physical Line multicomputer topology. The Ring topology is also equ ...
Some descriptive set theory 1 Polish spaces August 13, 2008
... dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X is a Polish space if X is a separable topological space that is metrizable by ...
... dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X is a Polish space if X is a separable topological space that is metrizable by ...
Metrics in locally compact groups
... last two results imply that the metric p is compatible with the topology of G and so the lemma is proved. It is clear that the above metric p will not yield bounded spheres for all radii unless, perhaps, G is compact, since G is necessarily covered by each (open) sphere of radius greater than 2M(Vl) ...
... last two results imply that the metric p is compatible with the topology of G and so the lemma is proved. It is clear that the above metric p will not yield bounded spheres for all radii unless, perhaps, G is compact, since G is necessarily covered by each (open) sphere of radius greater than 2M(Vl) ...
Measurable functionals on function spaces
... a topological space, the Baire sigma-algebra is the one generated by continuous real-valued functions. We are mainly interested in the following property of a dual pair of vector spaces (typically, E will be a space of signed measures and F will be a space of functions). DEFINITION. — Let E and F be ...
... a topological space, the Baire sigma-algebra is the one generated by continuous real-valued functions. We are mainly interested in the following property of a dual pair of vector spaces (typically, E will be a space of signed measures and F will be a space of functions). DEFINITION. — Let E and F be ...
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.