Question

If \(X\) and \(Y\) are topological spaces, \(\phi \in C(X, Y)\) is called proper if \(\phi^{-1}(K)\) is compact in \(X\) for every compact \(K \subset Y\). Suppose that \(X\) and \(Y\) are LCH spaces and \(X^{*}\) and \(Y^{*}\) are their one-point compactifications. If \(\phi \in C(X, Y)\), then \(\phi\) is proper iff \(\phi\) extends continuously to a map from \(X^{*}\) to \(Y^{*}\) by setting \(\phi(\infty, X)=\infty_{Y}\).

Short answer

The map \(\phi\) is proper if and only if it extends continuously to the one-point compactifications, mapping \(\infty_X\) to \(\infty_Y\).

Step-by-step solution

Understanding Proper Maps

A function \(\phi: X \rightarrow Y\) between topological spaces is called proper if for every compact subset \(K\) in \(Y\), the pre-image \(\phi^{-1}(K)\) is compact in \(X\). This definition will be used to verify if \(\phi\) extends continuously to the one-point compactifications \(X^*\) and \(Y^*\).

One-point Compactification

Since \(X\) and \(Y\) are locally compact Hausdorff (LCH) spaces, their one-point compactifications, \(X^*\) and \(Y^*\), respectively, extend the original spaces by adding a point at infinity \(\infty_X\) and \(\infty_Y\). These one-point compactifications are used to check if \(\phi\) can be continuously extended.

Proper Map Condition

For \(\phi\) to be proper, we require \(\phi^{-1}(K)\) to be compact for every compact \(K \subset Y\). This means \(\phi\) behaves well at compact parts, and its behavior at infinity is what influences continuation to \(X^*\).

Extension to One-Point Compactification

The statement is equivalent; \(\phi\) is proper if it can be extended to a continuous function, mapping \(\infty_X\) to \(\infty_Y\), on the one-point compactifications. This ensures that \(\phi\) respects the structure of both spaces at infinity.

Checking Continuity at Infinity

The requirement for \(\phi\) to be continuous at \(\infty_X\) ensuring it maps to \(\infty_Y\) is justified by compactness. For a proper map, this continuity holds, meaning \(\phi\) respects the compact nature even at the added infinity point.

Key concepts

Proper Maps

In topology, a function between two spaces is termed a "proper map" if it respects compact sets in a very specific way. This means if you start with a compact set in the target space and trace it back to the source space through the function, you'll end up with another compact set. Sounds straightforward? Here's the reason for its importance:

• A proper map ensures that the topology of the source space gets nicely "compactified," adhering to conditions that generally bolster continuity and comprehensibility.
• Proper maps matter because they maintain "control" over the compactness, especially when dealing with complex spaces.

Understanding proper maps is vital as it tells us about how different mappings between topological spaces can be extended or modified under certain conditions.

One-Point Compactification

The one-point compactification is a fascinating concept where you add exactly one new point to your space, effectively "wrapping" it into a closed, bounded entity. Here's what you need to know:

• It converts a space that may have been infinite in extent, like a half-plane or an open disk, into a compact space by adding a single point at infinity.
• This process is useful as it permits a broader understanding of both the original and newly extended spaces.
• The one-point compactification is a helpful tool to test whether functions, like our \(\phi\), can be extended seamlessly within modified space contexts.

This concept facilitates examining the interaction and behavior of functions at the "edges" or "limits" of spaces.

Continuous Functions

In the world of topology, continuous functions play a central role, much like they do in calculus. A function \(\phi: X \to Y\) is continuous if small perturbations in \(X\) yield small shifts in \(Y\). But there's more to it than just intuition:

• Continuity implies that for every open set in \(Y\), the pre-image under the function is an open set in \(X\).
• This characteristic affirms that the function behaves predictably and consistently across the space.
• In the case of extending \(\phi\) to one-point compactifications, ensuring continuity at infinity keeps relationships tidy and stable.

An understanding of continuous functions is indispensable for proving such extensions and examining their limits.

Locally Compact Hausdorff Spaces

This class of spaces marries two significant properties—local compactness and the Hausdorff condition—to yield a very "nice" structure. Here’s what makes them crucial for our exercise:

• A space is locally compact if every point has a neighborhood basis consisting of compact sets.
• The Hausdorff property guarantees that any two distinct points can be separated by distinct open sets.
• Together, these properties ensure we can manage not just infinite expanses but also precise separations between points—key for properly extending maps like \(\phi\).

Locally compact Hausdorff spaces provide a friendly environment, allowing function behaviors to be examined clearly, even at infinite limits.