# One of These Days

When you cut something with a knife, does the blade come into contact with each of the little pieces? I think so.

Now consider a topological version of this question. Let $X$ be a connected topological space (with more than 1 point) which we “cut” by removing a point $a\in X$. Let $C$ be a connected component of $X\setminus \{a\}$; this is one of the little pieces. Is it true that $a\in \overline{C}$?

True if we replace “connected” by “path-connected”. Indeed, let $\gamma\colon [0,1]\to X$ be a continuous function such that $\gamma(0)\in C$ and $\gamma(1)=a$. If necessary, we truncate $\gamma$ so that $\gamma(t)\ne a$ for $0\le t<1$. Now the set $\gamma([0,1))$ is a path-connected subset of $X\setminus\{a\}$, hence $\gamma([0,1))\subset C$. Since $\gamma(t)\to a$ as $t\to 1-$, we have $a\in \overline{C}$.

False as stated, even if $X$ is a subset of $\mathbb R^2$. Here is a counterexample given by Niels Diepeveen on Math.StackExchange:

The set $X$ consists of the point $(0,0)$ and of closed line segments from $(1/n,0)$ to $a=(0,1)$. The union of line segments is connected, and since it is dense in $X$, $X$ is also connected. However, once $a$ is removed, each line segment becomes its own connected component, and so does the point $(0,0)$. Clearly, the closure of $\{(0,0)\}$ does not contain $a$.

In this example $X$ is not compact. It turns out that the statement is true when $X$ is a compact connected Hausdorff space (i.e., a continuum), but the proof (also given by Niels Diepeveen) is not easy.

## One thought on “One of These Days”

1. Dejan Govc says:

Have you heard of the Knaster-Kuratowski fan? It is a connected subset of the plane that becomes totally disconnected upon removal of a certain point. So the knife doesn’t touch any of the pieces that remain.