Is the compactification of a path-connected space path-connected? Why or why not? (I came across this question in my notes while studying for finals and I have no idea.)

The question title says it all. For the record, I have no reason to believe that this is true, but my question has a bit of a background. I am reading Ramanan’s Global Calculus book because I am interested in the isomorphism between singular cohomology with coefficients in a ring $R$ and sheaf cohomology with […]

Please just don’t present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \cdots\}$$ Well, we can see that: $$\frac{n+1}{n} = 1+\frac{1}{n} > 1$$ Therefore, $1$ is a lower bound for $A$, however I still need to […]

Theorem 2.43 Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof Since $P$ has limit points, $P$ must be infinite. Suppose $P$ is countable, and denote the points of $P$ by $\mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3}, \ldots$. We shall construct a sequence $\{V_{n}\}$ of neighborhoods as follows. Let $V_1$ be any neighborhood […]

I’m interested in characterizing closed maps in terms of nets. Since a map is closed iff $\overline{f(V)} \subseteq f(\overline{V})$ for all subsets $V$, I believe one possible such characterization is $f$ is closed iff for each net $x_\alpha$ such that $f(x_\alpha) \to y$, we have that $x_\alpha$ converges to the set $f^{-1}({y})$. By convergence to […]

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic topology is unfamiliar to me and I could not seem to find anything via google. I do know something about the $p$-adic topology though (cf this […]

In the answer to this question a free ultrafilter is shown to contain the cofinite filter. But does every free filter contain it too? Obviously the ultrafilter that extend it does.

My algebraic topology class is very bad at teaching, it just doesn’t explain what’s needed. Let me be specific, I am looking at this question, Q. Find the degree of $f_0 :S^1 \to S^1$ the constant map, such that $f_0: z \to 1$ Now, it spends a great deal of time and space on paper […]

I want to know the topological relation between a manifold and a simplicial complex. I know that a simplicial complex cannot be a manifold since its a union of simplices which are manifolds of distinct dimensions. Now can we say that every manifold can be given a structure of a simplicial complex? and more informally, […]

Prove: Every finite subset of $\mathbb{R}$ is closed definition of closed: A set $A$ is closed if it contains all it accumulation or limit points. definition of accumulation point: Let $A$ be a subset of $\mathbb{R}$. A point $p\in \mathbb{R}$ is an accumulation or limit point if and only if every open set $G$ containing […]

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