Investigate whether the quotient of a locally compact space is always locally compact | Step-by-Step Solution

TinyProf's Guide to Quotient Spaces and Local Compactness

What We're Solving: We need to determine whether quotient spaces preserve local compactness - that is, if we start with a locally compact space and create a quotient space, will the result always be locally compact?

The Approach: This is a great example of mathematical reasoning! We'll examine the general principles, analyze why the initial reasoning seems plausible, then carefully check the counterexample to see what's really happening.

Step-by-Step Solution:

Step 1: Understand the Initial Reasoning The reasoning provided has some truth to it:

• ✓ Continuous maps do preserve compactness (compact sets map to compact sets)
• ✗ However, "the continuous image of a basis is a basis" is not generally true!

This is the key flaw. A basis for the domain doesn't necessarily give us a basis for the codomain under a continuous map.

Step 2: Recall What Local Compactness Means A space X is locally compact if every point has a neighborhood base consisting of compact sets. Equivalently, every point has a compact neighborhood.

Step 3: Analyze the Counterexample ℝ/ℤ The circle ℝ/ℤ is homeomorphic to the unit circle S¹, which is compact and locally compact.

Step 4: Find a True Counterexample Here's a classic counterexample:

• Take X = ℝ² (which is locally compact)
• Consider the equivalence relation that identifies all points on each line y = n (for integer n)
• The quotient space X/~ is not locally compact

Why? In the quotient, the "limit lines" create points that cannot have compact neighborhoods.

Step 5: A Cleaner Counterexample An even simpler example:

• Take the locally compact space ℝ
• Consider the quotient map that sends ℝ to ℝ ∪ {∞} but with the indiscrete topology on the image
• This quotient is not locally compact because no point has a compact neighborhood in the indiscrete topology.

The Answer: No, quotients of locally compact spaces are not always locally compact. The initial reasoning fails because continuous images of bases aren't necessarily bases. While continuous maps preserve compactness, the quotient topology can destroy the local structure needed for local compactness.

Memory Tip: Remember: "Quotients can be quotient!" - meaning they can lose nice properties. Continuous maps preserve compactness globally, but quotient topology can mess up the local neighborhood structure that makes local compactness work.

Great job thinking through this problem and questioning the initial reasoning - that's exactly the kind of critical thinking that makes you a strong mathematician! 🌟