ALEXANDROFF ONE POINT COMPACTIFICATION PDF

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.

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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.

Embeddings into compact Hausdorff spaces may be of particular interest. Since every compact Hausdorff space is a Tychonoff space , and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.

The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact. A topological space has a Hausdorff compactification if and only if it is Tychonoff. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0,1]C, the space of all functions from C to [0,1].

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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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