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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Recommendations Alexandroff One Point Compactification In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact spaces and one point compactification are proved. The ordinal numbers. Formalized Mathematics, 1 1 , The "way-below" relation.

Formalized Mathematics, 6 1 , Bases and refinements of topologies. Formalized Mathematics, 7 1 , Functions and their basic properties. Some basic properties of sets.

Compact spaces. Formalized Mathematics, 1 2 , Families of subsets, subspaces and mappings in topological spaces. Finite sets. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2 5 , On nowhere and everywhere dense subspaces of topological spaces. Formalized Mathematics, 4 1 , Introduction to meet-continuous topological lattices.

Formalized Mathematics, 7 2 , Locally connected spaces. Formalized Mathematics, 2 1 , Topological spaces and continuous functions. Domains and their Cartesian products.

Tarski Grothendieck set theory. Properties of subsets. Relations and their basic properties. Relations defined on sets.

Subsets of topological spaces.

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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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