Let $X$ be a compact Hausdorff space such that for every $x \in X$ , there exist a nested sequence of open sets $\{U_n\}$ such that $\{x\}=\bigcap_{n=1}^\infty U_n$ , then is it true that $X$ is first countable?

A topology is closed under arbitrary union of open sets, and therefore a fortiori also closed under countable union of open sets. Why do we require arbitrary union rather than just countable union? What might be an example that would illustrate why we require arbitrary union? Or are they just equivalent when talking about unions […]

I am reading the paper “Calculating the fundamental group of an orbit space” by M A Armstrong where he states the following – Let $\mathbb R$ act on the torus $T\cong S^1\times S^1$ by $$r\cdot(e^{2\pi ix},e^{2\pi iy})=(e^{2\pi i(x+r)},e^{2\pi i(y+r\sqrt{2})})$$ This action lifts to an action of $G=\pi_1(T)\times\mathbb R$ on $\mathbb R^2$ which has the same […]

(1) In the second example in Section 3.1 of the Wikipedia article on filter, the last sentence says: A nonprincipal filter on an infinite set is not necessarily free. On the other hand, Martin Sleziak’s answer to this question (and also the second sentence of this question) says: Filter, which is not free is called […]

Take $X = [0,1]$, and a continuous map $f:X \rightarrow X$. Then there exist a point $x \in X$ s.t. $f(x) = x$. We may take $X = (0,1)$ or $X = (0,1]$. Shall we get such fixed points in latter cases? Why or why not? Please explain. Thanks.

This question already has an answer here: If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval? 1 answer

I have the following question. Let $M$ be a smooth manifold which is homeomorphic to $\mathbb{R}P^{2}$. If one cuts $M$ along a non-contractible path then $M$ should be homeomorphic to a closed disc, right? Why is this so? Ben

Let $f$ be a continuous closed function from $X$ to $Y$ where $X$ and $Y$ are topological spaces. (Closed means that for any closed set $C$, $f(C)$ is also closed). Suppose that for any $y$ in $Y$, the inverse image of $y$ is compact. Show that if $K$ is a compact subset of $Y$, then […]

This sounds obvious (all balls of all rational radii on rational centres should work, I think) but I can’t complete the proof simply from these two properties of $\mathbb R^n$. $\mathbb R^n$ has a basis of open balls (all balls of all radii on all centres) $\mathbb R^n$ is second countable and Lindelöf So from […]

Could someone please help me in understanding the concepts of topologies and equivalent metrics. If possible, giving some examples of equivalent metrics. For example, I don’t know why for the Euclidean space, the d1, d2 and d(infinity) metrics are (strongly) equivalent. I would really appreciate any help! Thanks 🙂

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