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 […]

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) […]

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.

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 $ […]

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 […]

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

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 […]

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 […]

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 […]

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?

Intereting Posts

How to justify small angle approximation for cosine
Harmonic functions on $\mathbf{Z}^2$
Painting the faces of a cube with distinct colours
Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform random variables.
Prove $e^x – e^y \leq e |x-y|$ for $x$ belonging to $$
What's the relationship between a measure space and a metric space?
Probability distribution function that does not have a density function
When can you switch the order of limits?
Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
Confusion related to context sensitive grammar creation
max and min versus sup and inf
Functions that are their own Fourier transformation
Topology on $R((t))$, why is it always the same?
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
Banach spaces and their unit sphere