Articles of general topology

Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.

As the title says, I’m trying to prove that if $X/R$ is a Hausdorff space then $R\subset X\times X$ is closed. I have several questions about this: $(1)$ What exactly is $R$? I thought of $R$ as an equivalence relation, but I never thought of that relation (or any) as a set. What are the […]

Fubini's theorem on a product of locally compact spaces which do not have countable bases

Let $X$ be a locally compact Hausdorff space. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open subsets of $X$. A measure $\mu$ on $\mathcal B$ is called a (positive) Radon measure if it satisfies the following conditions. 1) $\mu(K) \lt \infty$ for every compact set $K$. 2) $\mu(U) = sup \{\mu(K) […]

A question about dimension and connectedness in order topologies

Does there exist a totally disconnected topological space T whose topology is the order topology of a linearly ordered set and whose (small inductive) dimension is equal to 1? There exist topological spaces which satisfy all the other conditions, but I do not know of any whose topology is an order topology.

Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy sequence in $ X $ is a Cauchy sequence in $ […]

Show that $ \cap \mathbb{Q}$ is not compact in $$

The title said it all. I have come up with a solution, but I cannot figure out some details. Please help me out and comment on my solution. Feel free to leave your own solution so that I can also learn from you. Show that $[0,1] \cap \mathbb{Q}$ is not compact in $[0,1]$. Solution: Let […]

Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$

This question already has an answer here: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ 6 answers

Proving an operation of interior is a set of open sets.

I am looking at General Topology notes on Alternative Ways of Defining Topology and come up with this questions: If $\iota : \mathscr P (X) \to \mathscr P (X)$ is an operation of interior, then $\tau = \{A \in \mathscr P(X) \ : \ \iota (A) = A \}$ is a set of open sets […]

separation properties in Hausdorff, compact spaces

Suppose $X$ is a compact Hausdorff topological space, $C\subseteq X$ a closed subset and $x\notin C$ a point. I have to prove that there exists a compact neighborhood of $x$ which is disjoint from $C$. Here’s what I did: since $X$ is Hausdorff, for every $y\in C$ we have two disjoint neighbourhoods of $y$ and […]

continuous monotonic function

How to prove that strictly monotonic continuous function carries open intervals to open intervals? if $f:\mathbb{R}\to \mathbb{R}$ continuous and monotonic, we need to to Prove that If $X \subset$ R is an open interval then $f(X)$ is an open interval. Clearly $f(X)$ is an interval(as continuous functions takes connected sets to connected sets and connected […]

What (and how many) pieces does the Banach-Tarski Paradox break a sphere into?

Wikipedia says There exists a decomposition of the ball into a finite number of non-overlapping pieces which can then be put back together in a different way to yield two identical copies of the original ball. So my question is, how many pieces is the ball broken into, and what are their shapes like?