Investigate whether an open set in R^m can be considered closed when viewed in a higher-dimensional space R^n | Step-by-Step Solution

Let's Explore Open Sets Across Different Dimensions! 📐

What We're Solving:

We want to understand what happens when we take an open set that lives in R^m and view it as a subset of a higher-dimensional space R^n (where n > m). Specifically, can such a set ever be closed in the larger space?

The Approach:

This is a beautiful question about how context matters in topology! We need to understand:

• How openness and closedness are defined relative to the ambient space
• What it means to "embed" a lower-dimensional space into a higher one
• Whether the topological properties change when we change our perspective

The key insight is that open and closed are relative concepts - they depend entirely on what space we're considering as our "universe."

Step-by-Step Solution:

Step 1: Clarify the Setup When we talk about an open set in R^m being viewed in R^n, we typically mean we're looking at R^m as a subset of R^n. For instance, we might view R² as the subset {(x,y,0) : x,y ∈ R} inside R³.

Step 2: Recall the Definitions

• A set U is open in a space X if every point in U has a neighborhood entirely contained in U
• A set C is closed in a space X if its complement X\C is open in X
• These definitions depend crucially on what space X we're working in!

Step 3: Consider What Happens to R^m in R^n Here's the crucial observation: when we view R^m as a subset of R^n (n > m), the entire space R^m has empty interior in R^n.

Why? Because any open ball in R^n around a point in R^m will contain points that stick out in the extra n-m dimensions, so they won't be in R^m.

Step 4: Analyze Open Sets Take any non-empty open set U in R^m. When viewed as a subset of R^n:

• U cannot be open in R^n (since it has empty interior, just like all of R^m)
• But could U be closed in R^n?

Step 5: Test with a Specific Example Let's consider the open unit disk U = {(x,y) : x² + y² < 1} in R², viewed as a subset of R³.

Is U closed in R³? A set is closed if and only if it contains all its limit points. Consider the sequence of points: (1/2, 1/2, 1/n) for n = 1, 2, 3, ...

This sequence approaches (1/2, 1/2, 0) ∈ U, but the sequence points themselves are not in U (they're not even in the R² plane). So U is not closed in R³.

The Answer:

No! An open set in R^m cannot be closed when viewed in R^n (n > m), assuming we embed R^m in the standard way.

Here's why: Any non-empty open set U in R^m will have boundary points in R^m that remain boundary points in R^n, but we can always find sequences in R^n that approach points in U from "outside" the R^m subspace. This means U cannot contain all its limit points in R^n, so it cannot be closed.

Memory Tip:

Think of it this way: "Going from a smaller room to a bigger house doesn't make your furniture more complete - there are still new ways to approach it!" When we move to higher dimensions, we create new directions from which points can approach our set, preventing it from being closed. 🏠✨

The beautiful lesson here is that topology is all about context - the same set can have completely different properties depending on which space we consider it to live in!