Some articles on metric space, space, metric, metric spaces, spaces:
Discrete Space - Properties
... The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the ... Thus, the different notions of discrete space are compatible with one another ... topology of a non-discrete uniform or metric space can be discrete an example is the metric space X = {1/n n = 1,2,3...} (with metric inherited from the real line and ...
... The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the ... Thus, the different notions of discrete space are compatible with one another ... topology of a non-discrete uniform or metric space can be discrete an example is the metric space X = {1/n n = 1,2,3...} (with metric inherited from the real line and ...
Compact Space - Other Forms of Compactness
... which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces ... Every real-valued continuous function on the space is bounded ... While all these conditions are equivalent for metric spaces, in general we have the following implications Compact spaces are countably compact ...
... which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces ... Every real-valued continuous function on the space is bounded ... While all these conditions are equivalent for metric spaces, in general we have the following implications Compact spaces are countably compact ...
Pseudocompact Space - Properties Related To Pseudocompactness
... In order that a Tychonoff space X is pseudocompact it is necessary and sufficient that every locally finite collection of non-empty open sets of X is finite ... Every countably compact space is pseudocompact ... For normal Hausdorff spaces the converse is true ...
... In order that a Tychonoff space X is pseudocompact it is necessary and sufficient that every locally finite collection of non-empty open sets of X is finite ... Every countably compact space is pseudocompact ... For normal Hausdorff spaces the converse is true ...
Category Of Metric Spaces
... In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric ... This is a category because the composition of two metric maps is again a metric map ... The monomorphisms in Met are the injective metric maps, maps that do not map two points into a single point ...
... In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric ... This is a category because the composition of two metric maps is again a metric map ... The monomorphisms in Met are the injective metric maps, maps that do not map two points into a single point ...
Types of Metric Spaces - Compact Spaces
... A metric space M is compact if every sequence in M has a subsequence that converges to a point in M ... This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable ... Examples of compact metric spaces include the closed interval with the absolute value metric, all metric spaces with finitely many points, and the Cantor set ...
... A metric space M is compact if every sequence in M has a subsequence that converges to a point in M ... This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable ... Examples of compact metric spaces include the closed interval with the absolute value metric, all metric spaces with finitely many points, and the Cantor set ...
Famous quotes containing the word spaces:
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)
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