Articles of general topology

Is such an infinite dimensional metric space, weakly contractible?

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 […]

Product of Two Metrizable Spaces

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 […]

Is every locally compact Hausdorff space paracompact?

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.

Compact space T2 satisfies first axiom of countability

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 $x$.

Prove that Euclidean, spherical, and hyperbolic 3-manifolds are THE only three-dimensional geometries that are both homogeneous and isotropic.

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 […]

On eventually constant sequences

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.

Definition of uniform structure

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 […]

Equivalent distances define same topology

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 […]

$X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, show that $d(a,b) = d(K,F)$ for some $a$, $b$

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 […]

One point compactification is second countable

$(X,\tau)$ 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 […]