Discussion
Consider the following statements:
P: The family of subsets \(\left\{A_{n} = \left(-\frac{1}{n}, \frac{1}{n}\right), n = 1, 2, \ldots\right\}\) satisfies the finite intersection property.
Q: On an infinite set \(X\), a metric \(d : X \times X \to R\) is defined as \(d(x, y) = \begin{cases} 0, & x = y \\ 1, & x \neq y \end{cases}\). The metric space \((X, d)\) is compact.
R: In a Frechet (\(T_{1}\)) topological space, every finite set is closed.
S: If \(f : R \to X\) is continuous, where \(R\) is given the usual topology and \((X, \tau)\) is a Hausdorff (\(T_{2}\)) space, then \(f\) is a one-one function.
Which of the above statements are correct?
Answer: A
The correct statements are P and R.
P is true: the intervals \((-1/n, 1/n)\) are nested and all contain 0, so every finite subfamily has non-empty intersection. R is true: in a \(T_{1}\) space every singleton is closed, and a finite union of closed sets is closed.
Q is false: with the discrete metric every singleton is an open ball of radius \(1/2\), and these open sets cover the infinite set \(X\) with no finite subcover, so \((X, d)\) is not compact. S is false: a constant map from \(R\) into any Hausdorff space is continuous but not one-one. Any option containing Q or S is therefore wrong.
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