ALEXANDROFF ONE POINT COMPACTIFICATION PDF

References An example Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. The resulting compactification can be thought of as a circle which is compact as a closed and bounded subset of the Euclidean plane. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. Definition An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces , because of the special properties compact spaces have.

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Compactification (mathematics)

An example Edit Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. The resulting compactification can be thought of as a circle which is compact as a closed and bounded subset of the Euclidean plane. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below. Definition Edit An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces , because of the special properties compact spaces have.

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ALEXANDROFF ONE POINT COMPACTIFICATION PDF

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