Let X be Hausdorff and assume Y=X ∪{∞} is the one-point compactification of X. (a) Show that if

Step 1: Recall the definition of the oneu002Dpoint compactification. The oneu002Dpoint compactification

Let X be Hausdorff and assume Y=X ∪{∞} is the one-point...

Let $X$ be Hausdorff and assume $Y=X \cup\{\infty\}$ is the one-point compactification of $X$. (a) Show that if $X$ is not compact, then $\mathrm{Cl}(X)=Y$. (b) Show that if $X$ is compact, then $\mathrm{Cl}(X)=X$, and $Y$ is disconnected with $(\infty)$ being one of its components. (This shows that not much interesting happens when taking the one-point compactification of a space that is already compact.)

Let $X$ be Hausdorff and assume $Y=X \cup\{\infty\}$ is the one-point compactification of $X$.
(a) Show that if $X$ is not compact, then $\mathrm{Cl}(X)=Y$.
(b) Show that if $X$ is compact, then $\mathrm{Cl}(X)=X$, and $Y$ is disconnected with $(\infty)$ being one of its components. (This shows that not much interesting happens when taking the one-point compactification of a space that is already compact.)

Key Concepts

Connectedness and Disconnectedness

Connectedness is the characteristic of a space that cannot be partitioned into two disjoint nonempty open sets. When modifications like one-point compactification are applied, especially to compact spaces, the resulting space may become disconnected, revealing a richer structure in terms of its components and topological separation.

Closure of a Set

The closure of a set in a topological space is the smallest closed set that contains it. This includes all the limit points of the set as well as the set itself, and studying the closure helps in understanding how open and closed properties interact within compactifications and other topological operations.

One-Point Compactification

The one-point compactification is a technique used to convert a non-compact space into a compact space by adding a single, distinct point often referred to as 'infinity'. This method is particularly useful for studying spaces that are almost compact, as it alters the topology minimally while inducing the compactness property.

Compactness

Compactness is a property of a topological space where every open cover has a finite subcover. This concept is central to many areas of topology because it generalizes the notion of closed and bounded subsets in Euclidean spaces, and it is a key condition in determining how a space can be extended or compactified.

Hausdorff Spaces

A Hausdorff space is one in which any two distinct points have disjoint neighborhoods. This separation property is essential in ensuring the uniqueness of limits and plays a critical role in many topological constructions, including the one-point compactification, where the behavior of points at infinity must be carefully managed.

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