We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is […]

I am having trouble with a practice exam question: $$\text{Show that if $X$ and $Y$ are metrizable, then so is $X\times Y$}$$ What I have so far: Given metric spaces $(X,d_x)$ and $(Y,d_y)$, I know that since $d_x$ and $d_y$ are continuous, then $d_x \times d_y$ is also continuous, so all that I need to […]

It seems likely that for any open cover, we can construct a locally finite refinement using the local compactness of the space. I can’t figure out how to work the construction though, and I’m not yet convinced that there is no counterexample.

Let $(X,\tau)$ be a compact space $T_2$ that, for all $x \in X$, exists $\mathcal{U}_x \subseteq \tau$ countable that satisfies $\bigcap \mathcal{U}_x = \{x\}$. Prove that $(X,\tau)$ satisfies the first axiom of countability. I’m stuck in that problem because I don’t know how I find the countable base of $x$.

The book Introduction to Topology by C. Adams and R. Franzosa says : There are only three three-dimensional geometries that are both homogeneous and isotropic: the Euclidean, spherical, and hyperbolic geometries. It has been just written without giving even a little reason(s)/explanation, let alone a proof of it. What is an easiest rigorous proof for […]

It is of course true that in a discrete space a sequence converges iff it’s eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the space necessarily discrete? I want to examine this statement for metric spaces but use of Hausdorff spaces is always welcome.

From Wikipedia: A uniform space $(X, Φ) $is a set $X$ equipped with a nonempty family $Φ$ of subsets of the Cartesian product $X × X$ ($Φ$ is called the uniform structure or uniformity of $X$ and its elements entourages) that satisfies the following axioms: if $U$ is in $Φ$, then $U$ contains the diagonal […]

I have to prove that equivalent distances define same topology. I know there are similar questions, so please don’t have a go at me but I am still confused and they don’t answer it in the way I have been taught. If distances are equivalent then there exist an $\alpha$ and $\beta$ more than zero […]

Some caveats: Let $K$ be non-empty and compact, $F$ be non-empty and closed, $X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, where $d(x,y)$ is one of the metrics $d_1$, $d_2$, or $d_\infty$ on $\mathbb{R}^n$. Show that $d(a,b) = d(K,F)$ for some $a\in K$, $b\in F$. It seems […]

$(X,\tau)$ is a local compact, second countable Hausdorff space s.t. $ X = \cup_n K_n$ for countably many compact sets in $X$. Then $\infty$ has a countable neighborhood basis of open sets where $\infty \in (X_\infty,\tau_\infty)$. This is the one-point compactification of $X$. This is homework so I would appreciate some hints. I already saw […]

Intereting Posts

Algorithm for creating binary rational numbers
Real analyticity of a function
A statement about divisibility of relatively prime integers
The Adjacency Matrix of Symmetric Differences of any Subset of Faces has an Eigenvalue of $2$…?
Why rationalize the denominator?
Intuition — $c\mid a$ and $c\mid b$ if and only if $c\mid \gcd(a,b)$.
An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function
$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$
Infinite derivative
Algebraic numbers that cannot be expressed using integers and elementary functions
permutation group and cycle index question regarding peterson graph
Every infinite Hausdorff space has an infinite discrete subspace
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
Proof through combinatorial argument
What are spinors mathematically?