Discussion
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
\[ \Delta = \{(x, x, x, \cdots) \mid x \in \mathbb{R}\} \]
with the subspace topology induced from Y. Which of the following statements is TRUE?
Answer: B
The true statement is \(\Delta\) is locally compact.
The map x ↦ (x, x, x, ...) from \(\mathbb{R}\) onto \(\Delta\) is continuous, because each coordinate is continuous, and its inverse is the first projection, also continuous. So \(\Delta\) is homeomorphic to \(\mathbb{R}\), which is locally compact (and connected).
\(\Delta\) is not open in Y: every basic open set of the product topology restricts only finitely many coordinates, so it contains points whose coordinates are not all equal. It is not dense: \(\Delta\) is closed, being the set where every coordinate equals the first, and it is not all of Y. It is not disconnected, since \(\mathbb{R}\) is connected.
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