Articles of general topology

The integral of a closed form along a closed curve is proportional to its winding number

Source: Guillemin-Pollack Exercise 4.8.2. Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 – \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 – \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where $W(\gamma, 0)$ is the winding number of $\gamma$ with respect to the origin. $W(\gamma, 0)$ is defined just like $W_2(\gamma, 0)$, but using […]

Are all Infinite Simplicial Complexes non-compact?

I recently saw a proof of this using the fact that the star of a vertex $v$ of a simplicial complex is open. However, this does not hold if $st(v) = v$ where $st(v)$ means the star of $v$ (i.e. $v$ lies only in a zero-simplex). Is there any reason as to why $ K […]

A generalization of the connected sum of links

A connected sum of two links $K$ and $L$ involves cutting a segment in each link and joining them up as illustrated in the top diagram, the connected sum of two trefoil knots. Is there any terminology for taking a link and doing the same process but to itself, as illustrated by the bottom diagram, […]

Consider the “infinite broom”

Consider the “infinite broom” $X$ pictured in figure below. Show that $X$ is not locally connected at $p$, but is weakly connected at $p$.[Hint: Any connected neighborhood of $p$ must contain all the points $a_i$] For simplicity, I took, $X\subset \mathbb{R}^2$, such that $p=0\times 0$ and $a_1=1\times 0$, so that consider the subspace topology of […]

Prove that $Ω$ has no accumulation point

Let $f,g,h,l:ℂ→ℝ$ four harmonic functions such that $f≠g$ and $h≠l$. Let $D$ be an open set in $ℂ$. Let us define the set: $$Ω:\{ s=α+iβ∈D:f(s)=g(s),h(s)=l(s)\}$$ My question is: Prove that $Ω$ has no accumulation point, i.e., the set $Ω$ is discrete.

Proving that a compact subset of a Hausdorff space is closed

I am having trouble understanding the answers here. I am trying to prove that a compact subset of a Hausdorff space is closed. Following the proof is difficult, perhaps because Brian reused letters for different things(although I get they are arbitrary, I can’t follow it.) The second answer uses nets and filters, which I don’t […]

Are all paths with the same endpoints homotopic in a simply connected region?

It is clear to me that if all paths (with the same endpoints) in a region are homotopic then that region is simply connected, however I am having difficultly proving the converse, that is, all paths with the same endpoints are homotopic in a simply connected region. Here is what I have so far Given […]

Intersection of closure of decreasing family of sets

Suppose we have a countably infinite family of non-empty, bounded open sets, $A_{0},A_{1},\dots$ in $\mathbb{R}^n$ such that $A_{i}\supseteq A_{i+1}$ for all $i$. If the intersection $$A=\bigcap_{i=0}^{\infty}A_{i}$$ is non-empty and closed, is it true that we have $$\bar A=\bigcap_{i=0}^{\infty}\bar{A_{i}}$$ where $\bar X$ denotes the closure of $X$? I think it is true intuitively, but I can […]

Does perfectly normal $\implies$ normal?

A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets $E$ and $F$ can be precisely separated by a continuous function $f$ from $X$ to the […]

A sequentially compact subset of $\Bbb R^n$ is closed and bounded

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence. Let $U$ be a subset of $\mathbb{R}^n$ such that $U$ is not closed. Construct a […]