Some examples concerning proximinality in Banach spaces

We discuss the relationships between ττ-strongly Chebyshev, ττ-approximative compactness, ττ-strong proximinality, and proximinality, where ττ stands for the norm topology or weak topology. We consider four examples which illustrate the differences, and a counterexample that shows the question regarding the approximative nn-compact Chebyshev subset, raised by Bandyopadhyay et al., is negative.