Articles of general topology

Constructing a local nested base at a point in a first-countable space

I am trying to prove the following: Let $X$ be a first countable space and $x$ a member of $X$. Prove that there is a local nested basis $\{S_n\}_{n=1}^\infty$ at $x$. Since $X$ is first countable there is a countable local base $\mathcal{B}_x$ at $x$. Constructing a nested sequence of subsets of $\mathcal{B}_x$ is easy. […]

Convergence/Sequences/Box Topology

Let $\mathbb{R}^\omega$ be the countable product of $\mathbb{R}$. Make it a topological space using the box topology. Show that the sequence $\{(1/n, 1/n, ….)$ | $n \in \mathbb{Z}_+\}$ does not converge to $(0,0,…)$. I know that a given sequence $\{x_n$ | $n \in \mathbb{Z}_+\}$ such that each $x_n$ is in $X$, $\{x_n\}$ converges to x […]

Are these two definitions equivalent?

If $\langle A_n : n \in \omega \rangle$ is a sequence of subsets of a set $X$, $$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $$ If $\mathcal A$ is a family of subsets of a set $X$, then, $L(\mathcal A)$ denotes […]

Compatibility of topologies and metrics on the Hilbert cube

Consider the Hilbert cube $Y = [0,1]^\mathbb{N}$. It is easy to define four classes of metrics on $Y$ for $\gamma>0$ and $\omega>1$: $$d^\gamma_{sup,pol}(x,y) = \sup_{k\geq 1} |x_k-y_k|/k^\gamma,$$ $$d^{\gamma}_{sum,pol}(x,y) = \sum_{k=1}^\infty |x_k-y_k|/k^{1+\gamma}, $$ $$d^\omega_{sup,exp}(x,y) = \sup_{k\geq 1} |x_k-y_k|/\omega^k,$$ $$d^\omega_{sum,exp}(x,y) = \sum_{k=1}^\infty |x_k-y_k|/\omega^k $$ Question, do all these metrics define the same topology? does any of this […]

Equivalence of continuous and sequential continuous implies first-countable?

It is an immediate result that a map from a first-countable space is continuous iff it is sequentially continuous. I was wondering if the converse was also true. That is, is it true that if every map from a space $X$ is continuous iff it is sequentially continuous, then $X$ must be first-countable?

Longest sequence of minimally finer topologies

Suppose we start with a topology $T_1$ of X. Is there a way to get construct a sequence of topologies $T_n$ such that $T_{n – 1} \subset T_{n}$ in which there is no finer topologies in between, also that sequence is the longest one‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌.

Proof of non-existence of a continuous bijection between $\mathbb{R}$ and $\mathbb{R}^2$

There are a lot of websites and forums, which explain that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and even give some bijections. (By the way: Can you generalize it? since it works with the natural Numbers and the real numbers, does there exist a bijection between any infinite set $X$ and $X^2$) On […]

Topological Conditions Equivalent to “Very Disconnected”

Definition: Let $(X, \mathcal{T})$ be a topological space, where the set $X$ has more than one element. Suppose that for every pair of distinct elements $a, b \in X$, there exists a separation $(A,B)$ of $X$ such that $a \in A$ and $b \in B$. Then we say $(X, \mathcal{T})$ is very disconnected. Is this […]

Separable implies second countable

We have $(X,d)$ a metric space. The problem I want to prove is quite long so I’ll just put what I need to get it: if $X$ is compact then is separable if $X$ is separable then is second countable I’ve already proved the first one, but I’m having a trouble trying to prove the […]

Is there a set of all topological spaces?

This question is from Willard’s General Topology: Is there a set of all topological spaces? My try is: Suppose $\mathfrak T $ is set of all topological spaces, then $\mathfrak T $ ‘contains’ all the sets (i.e., if $S$ is some set, then $\{\varnothing, S\}\in\mathfrak T $). Since Willard assumes that a set cannot be […]