Articles of general topology

If $U \subset S \subset X$, $U$ is open in $S$, and $S$ is open in $X$, then $U$ is open in $X$.

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 […]

Show that $\lbrace x_n : n \in \mathbb{N} \rbrace \cup \lbrace x \rbrace$ is a compact subset of $(X,d)$

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 […]

Showing that that the torus is not a retract of a $3$-sphere

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.

Do we need net refinements not induced by preorder morphisms?

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' […]

What is the topological dimension of the Peano curve?

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 […]

Path-connectedness and compactifications

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.)

Are locally contractible spaces hereditarily paracompact?

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 […]

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

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 […]

Problems in Theorem 2.43 of baby Rudin

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 […]

Characterization of closed map by sequences/nets

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 […]