Question

Set \( X_n := \mathbb{R} \) for each \( n \in \mathbb{N} \). Define \( Y := \prod_{n \in \mathbb{N}} X_n \). Endow \( Y \) with the product topology, where the topology on each \( X_n \) is the Euclidean topology. Consider the set \[ \Delta = \{ (x, x, x, \dots) \mid x \in \mathbb{R} \} \] with the subspace topology induced from \( Y \). Which of the following statements is TRUE?

• A. \( \Delta \) is open in \( Y \)
• B. \( \Delta \) is locally compact
• C. \( \Delta \) is dense in \( Y \)
• D. \( \Delta \) is disconnected

Hint:

A subspace of a locally compact space is locally compact if the subspace itself is homeomorphic to a locally compact space. In this case, \( \Delta \) is homeomorphic to \( \mathbb{R} \), which is locally compact.

Answer 1

The Correct Option is B

We are given the set \( Y := \prod_{n \in \mathbb{N}} X_n \), where each \( X_n \) is \( \mathbb{R} \) with the Euclidean topology, and \( \Delta \) is the set of points in \( Y \) where all coordinates are the same, i.e., the diagonal in \( Y \). We need to analyze the properties of \( \Delta \) with respect to the subspace topology induced from the product topology on \( Y \).
Step 1: Open set analysis.
In the product topology, a basic open set in \( Y \) is of the form \( \prod_{n \in \mathbb{N}} U_n \), where \( U_n \) is open in \( X_n = \mathbb{R} \) and \( U_n = \mathbb{R} \) for all but finitely many \( n \). Since \( \Delta \) consists of points where all coordinates are the same, it is not open in \( Y \) because for any point \( (x, x, x, \dots) \), any open neighborhood of that point will include points where the coordinates differ. Therefore, \( \Delta \) is not open in \( Y \), and option (A) is false.
Step 2: Compactness.
A set is locally compact if every point has a neighborhood base of compact sets. \( \Delta \) is homeomorphic to \( \mathbb{R} \) (via the map \( x \mapsto (x, x, x, \dots) \)), and since \( \mathbb{R} \) is locally compact, \( \Delta \) is also locally compact. Therefore, option (B) is true.
Step 3: Density.
For \( \Delta \) to be dense in \( Y \), every open set in \( Y \) must intersect \( \Delta \). However, since \( \Delta \) consists only of points where all coordinates are equal, it does not intersect every open set in \( Y \). Therefore, \( \Delta \) is not dense in \( Y \), and option (C) is false.
Step 4: Connectivity.
\( \Delta \) is homeomorphic to \( \mathbb{R} \), which is connected. Thus, \( \Delta \) is connected, and option (D) is false.
Step 5: Final Answer.
The correct answer is (B) because \( \Delta \) is locally compact.