Some word to say 1. “Boundedness” is not a topological property. In fact, in a general topological space, we cannot tell what is mean by “bounded”. 2. Second, the Bolzano-Weierstrass property may need to be modified. The concept “sequence” need to be generalized to “net”. However, the concept “subnet” of a “net” is not exactly the same as the concpet “subsequence” of a “sequence”. According to John Kelley, the term subnet is defined as follow: Let X be a topological space. A net f: D->X is a map defined on a directed system D into X. A net g:E->X is said to a subnet of the net f is there exists a map T: E->D which satisfies: 1. g=foT 2. T maps the “tail” of E into the “tail” of D in the following sense For each s in D, there exists t in E such that T(u)>s whenever u>t. However, we do not require T to be “monotone” ! The requirement of “confinality” of the map T should be avoid as it is not suitable for all purposes. (See General Topology, John Kelley, P.70) Therefore, if we regard a sequence as a net, a subnet of this net may NOT be a subsequence of the original sequence in the general sence of analysis. Hence, I doubt the statement “A sequence in a compact space has a converging subsequence.”

if the term “subsequence” requires confinality.