Suppose $S$ is a subspace of the topological space $X$. If $U \subset S \subset X$, $U$ is open in $S$, and $S$ is open in $X$, then $U$ is open in $X$. Same is true with “closed” instead of “open”. I’ve solved the open version, but for closed version I get a this expression […]

Let $(X,d)$ be a metric space, and $(x_n)_{n \in \mathbb{N}}$ a sequence in $X$, with $x_n \rightarrow x \in X$ (w.r.t the usual metric $d$). Show that $\lbrace x_n : n \in \mathbb{N} \rbrace \cup \lbrace x \rbrace$ is a compact subset of $(X,d)$ Need step by step proof! These are my thoughts: I know […]

My question is obviously based on the title. I want to show that there is no retraction of a $3$-sphere (denoted $S^3$) onto the torus $T^2$ (doughnut surface). Any ideas on how one should do this? Input would be highly appreciated.

From Engelking’s book on general topology (slightly rephrased): Definition: We say that the net $S': \Sigma' \to X$ is finer than the net $S: \Sigma \to X$ if 1. there exists a function $f: \Sigma' \to \Sigma$ such that for any $\sigma'_0 \in \Sigma'$ there exists a $\sigma_0 \in \Sigma$ such that for any $\sigma' […]

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it’s on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension. So one can […]

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 […]

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