Does there exist a compact topological space not having countable basis? I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. Tychonoff’s Theorem shows that this topological space is compact Hausdorff space, but I’m not sure how to prove that this space does not have any countable basis. […]

Given $(X, \mathcal{T})$ a topological space. Let $\mathcal{S}$ be a subbasis on $(X, \mathcal{T})$ Claim: If $\mathcal{S}$ is countable, then $\mathcal{T}$ has a countable basis $\mathcal{B}$ I am not sure how to go about approaching this quetion but here’s my attempt: I want to show that there exists a surjection $g$ from $\mathcal{S}$ to $\mathcal{B}$, […]

In topological space, does first countable+ separable imply second countable? If not, any counterexample?

Another question about the convergence notes by Dr. Pete Clark: http://math.uga.edu/~pete/convergence.pdf (I’m almost at the filters chapter! Getting very excited now!) On page 15, Proposition 4.6 states that for the following three properties of a topological space $X$, $(i)$ $X$ has a countable base. $(ii)$ $X$ is separable. $(iii)$ $X$ is Lindelof (every open cover […]

Let X be a compact metrizable space. Would you help me to prove that X has a countable basis. Thanks.

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I’m looking for a similar criterion for locally compact Hausdorff spaces, using $C_0(X)$ (the Banach space of continuous functions vanishing at infinity) instead of $C(X)$. Naive Guess: Suppose $X$ is a […]

How do I prove that a metric space is separable iff it is second countable?

Background: This question came up in my homework (but was not a homework problem). The problem was proving one direction of the Heine-Borel theorem. As with all proofs of compactness, one begins with, “Suppose $A$ is closed and bounded, and $\mathcal{U}$ is an open cover …” My proof, which I believe is typical, constructed a […]

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