### general topology - Are closed subsets of a locally compact space, locally compact?

I know that there are open subsets of locally compact topological spaces that are not locally compact ($\mathbb{Q}$ in the Alexandroff's compactification).I wonder if any closed subset of a locally compact space is always locally compact.Definition. $(X,\tau)$ is locally compact if for each $x\in X$, there is a neighborhood of $x, A_x\in \tau:closure(A_x)$ is compact....Read more

### functional analysis - what exactly is weak* topology?

I know that weak* topology is the weakest topology so that $Jx$ is continuous for $\forall x\in X$, where $J$ is the isometry from $X$ to $X''$. But what exactly is this topology? What is the open set in general look like?And moreover, I want to prove another topology induced by metric is exactly the weak* topology. How can I prove this topology is weaker than weak* topology, so that it is exactly the weak* topology?...Read more

### Subbasis of a weak topology

Let $\{(X_\alpha,\mathscr{T}_\alpha):\alpha\in\Lambda\}$ be an indexed family of topological spaces, and for each $\alpha\in\Lambda$ let $f_\alpha:X\to X_\alpha$ be a function. Furthermore, let $\mathscr{T}$ be the weak topology on X induced by $\{f_\alpha:\alpha\in\Lambda\}$. Then $\mathscr{L}=\{f_\alpha^{-1}(U_\alpha):\alpha\in\Lambda; U_\alpha\in\mathscr{T}_\alpha\}$ is a subbasis for $\mathscr{T}$.I have tried this: Since $f_\alpha$ is continuous therefore $f_\alpha^{-1}(U_\alpha)$ is open then, intersections of elements of $\mathscr{L}$ ar...Read more

### general topology - In proving every open connected subset of $\mathbb{R}^n$ is path connected

I would like to prove that every open connected subset of $\mathbb{R}^n$ is path connected.Let us choose $E$ to be a such open connected subset, then given any point $p \in E$, we will define $F$ be the set of all points in E that can be joined to $p$ by a path in $E$.The idea is to show that $F = E$ by showing $F$ is clopen as well as $F$ is path connected.We choose $q \in F \subseteq E$, then since $E$ is open, we can find an open ball $D_x(q,\epsilon)$ in $E$.Here are my questions.My lecturer said $D_x(q,\epsilon)$ is path connected as any p...Read more

### general topology - Compactness of a metric space

If a metric space $(X,d)$ is compact then for every equivalent metric $\sigma$, $(X,\sigma)$ is complete. This is because, for any cauchy sequence in $(X,\sigma)$ has a convergent subsequence due to fact $(X,\sigma)$ is a compact metric space, hence original sequence is convergent. My question is , does the converse also hold ? That is, let $(X,d)$ be a metric space such that for every equivalent metric $\sigma$, $(X,\sigma)$ is complete. Does this imply $(X,d)$ is compact?...Read more

### general topology - Totally bounded, sequentially compact, complete, bounded, closed, equicontinuous $\Rightarrow$ compact?

Related; When $K$ is compact, if $S\subset C_b(K)$ is closed,bounded and equicontinuous, then $S$ is compact? (ZF)I just edited my whole question since i think it was a bit messy.Here is my question.Let $K$ be a separable compact metric space and $S\subset C(K,\mathbb{C})$.Let $S$ be closed,bounded,uniformly equicontinuous on $K$, sequentially compact, totally bounded and complete.Then is $S$ compact? (in ZF)Thank you in advance!...Read more