Articles of general topology

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 […]

How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?

This question already has an answer here: Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate] 2 answers $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected? 3 answers

Alternative definition of covering spaces.

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and $B$ is a covering map (and $E$ a covering space of $B$) if for every $b\in B$ there is […]

Show that a set is closed and open

We say that $f: X \to M$ is bounded if $f(X) \subset B_r(a) = \{m \in M \mid d(m,a) \lt r\}$ for some $r \in \mathbb{R}^+$ and $a \in M$. (NOTE: $(M,d)$ is a metric space with a distance $d$). We also have a distance on $M^X$: $$\hat{d}(f,g) = \begin{cases} 1, &\text{if } \exists x […]

Relative compactness of metric space

I know that in a metric space $X$ compactness, countable compactness and sequential compactness of a subspace $X’$ are equivalent using the definition of countable compactness as every infinite subset of $X’$ has an accumulation point in $X’$ and of sequential compactness as every sequence in $X’$ has a subsequence converging to a point in […]

Affine transformation

Let $S_1$ and $S_2$ be sets. Let $n_1$ be the cardinality of $S_1$ and $n_2$ be the cardinality of $S_2$. I assume that $n_1$ and $n_2$ are finite. Let $e$ be a function that maps members of $S_1$ and $S_2$ to real numbers. Assume that we have: $(1/n_1) \sum\limits_{x_i \in S_1} e(x_i) \geq (1/n_2) \sum\limits_{x_i […]

(Baby Rudin) To show the set of all condensation points of a set in Euclidean space is perfect

This is from Rudin’s Principal of Mathematical Analysis, Chapter 2, Problem 27. Let $E \subseteq \mathbb{R}^k$. Let $P$ be the set of all condensation points of $E$. Let $\{ V_n \}$ be a countable base of $\mathbb{R}^k$. Let $W$ be the union of those $V_n$ for which $E \cap V_n$ is at most countable. Prove […]