# ‘flimsy’ spaces: removing any nn points results in disconnectedness

Consider the following property:

$$R\mathbb R$$ is a connected space, but $$R∖{p}\mathbb R\setminus \{p\}$$ is disconnected for every $$p∈Rp\in \mathbb R$$.

$$S1S^1$$ is a connected space and if we remove any point, it is still connected. But if we remove two arbitrary points $$pp$$ and $$qq$$, the resulting $$S1∖{p,q}S^1 \setminus \{p,q\}$$ is disconnected.

Let $$XX$$ be a topological space. Let’s call $$XX$$ to be $$nn$$-flimsy if removing fewer then $$nn$$ arbitrary points leaves the space connected and removing any $$nn$$ arbitrary (distinct) points disconnects the space.

We saw that $$R\mathbb R$$ is $$11$$-flimsy and $$S1S^1$$ is $$22$$-flimsy (as $$S1∖{∗}≅RS^1 \setminus \{*\} \cong \mathbb R$$).

Question: Is there a $$33$$-flimsy space?

So I’m searching for a space $$XX$$ such that the removal of any $$33$$ points disconnects the space, but fewer don’t.

I suspect that there is no such space. I thought I could show it by showing first, that $$11$$– or $$22$$-flimsy spaces are in some way unique, but I found many examples of $$11$$-flimsy spaces which are significantly different (the long line, a variant of the topological sinus, trees).

Alternatively: Is there a standard terminology for this property? (it definitely ‘feels’ like $$nn$$-connectivity in graph theory)

Addendum 1: A space $$X={x,y}X=\{x,y\}$$ with two points is a trivial $$33$$-flimsy example, since we cannot remove three distinct points. Of course, I’m interested in real examples.

Addendum 2: Since Qiaochu Yuan and Paul Frost argued that CW-complexes won’t work, here are some thoughts concerning the finite case:

Let $$(X,T)(X,T)$$ be a topological space with finite $$XX$$. Then $$TT$$ is automatically an Alexandrov topology and therefore has the Specialization preorder $$≺\prec$$.
If we have a connected component $$Z(x)Z(x)$$ of a point $$xx$$ in a finite space with Alexandrov topology, then $$Z(x)Z(x)$$ and its complement are closed and open, so they are downwardly closed. If we visualize $$(X,T)(X,T)$$ by the graph $$GG$$ which has $$XX$$ as vertices and two vertices $$v,wv,w$$ are connected if $$v≺wv\prec w$$ or $$w≺vw \prec v$$, then connected components in $$TT$$ refer to connected components of the graph. Deleting a point in $$XX$$ corresponds to deleting the respective vertex.

Claim: There is no finite $$11$$-flimsy space (disregarding the trivial examples above). Otherwise we have a graph where the removal of any vertex results in a disconnected graph. This graph can’t be finite.

Corollary: There are no finite $$nn$$-flimy spaces for $$n∈Nn\in \mathbb N$$ (disregarding the trivial examples above). The removal of one point results in a finite $$n−1n-1$$-flimsy space, which can’t exist (induction).

Still open: Are there nontrivial $$33$$-flimsy spaces? Those should be infinite and shouldn’t be homeomorphic to CW-complexes.

Addendum 3: Funfact: Every topological space can be embedded into a $$11$$-flimsy space. Just add a real line to each point (as a one-point union). Alternatively, add $$11$$-spheres to every point. Then add $$11$$-spheres to each new point. Continue like this for eternity.

Addendum 4: In the setting of Whyburn’s book Analytic topology it is shown, that a compact set cannot be $$11$$-flimsy (Chapter 3, Theorem 6.1). Since all my examples for $$11$$-flimsy spaces are non-compact: Is there an example of a compact $$11$$-flimsy space? Are all $$nn$$-flimsy spaces non-compact (at least they are infinite)?

First, we show that if $$XX$$ is a 2-flimsy space and $$x≠y∈Xx\neq y\in X$$, then $$X∖{x,y}X\backslash\{x,y\}$$ has exactly two connected components. For this, we consider 3 open sets $$U1,U2,U3U_1,U_2,U_3$$ such that $$(U1∪U2∪U3)∩{x,y}c=X∖{x,y}(U_1\cup U_2\cup U_3)\cap\{x,y\}^{c}=X\backslash\{x,y\}$$, $$U1∩U2∩{x,y}c=U1∩U3∩{x,y}c=U2∩U3∩{x,y}c=∅U_1\cap U_2\cap\{x,y\}^{c}=U_1\cap U_3\cap\{x,y\}^{c}=U_2\cap U_3\cap\{x,y\}^{c}=\emptyset$$, and $$∀i∈{1,2,3}, Ui∩{x,y}c≠∅\forall i\in\{1,2,3\},\ U_i\cap\{x,y\}^{c}\neq\emptyset$$. If $$u1∈U1∩{x,y}cu_1\in U_1\cap\{x,y\}^{c}$$ and $$u2∈U2∩{x,y}cu_2\in U_2\cap\{x,y\}^{c}$$, then we can show $$X∖{u1,u2}X\backslash\{u_1,u_2\}$$ is connected.
The second big step is to consider $$x,t,s∈Xx,t,s\in X$$, three distinct points of a $$22$$-flimsy space. We denote $$C1(t),C2(t)C_1(t),C_2(t)$$ the two connected components of $$X∖{x,t}X\backslash\{x,t\}$$ and $$C1(s),C2(s)C_1(s),C_2(s)$$ the two connected components of $$X∖{x,s}X\backslash\{x,s\}$$. We suppose $$s∈C1(t)s\in C_1(t)$$ and $$t∈C1(s)t\in C_1(s)$$. Then $$D=C1(t)∩C1(s)D=C_1(t)\cap C_1(s)$$ is one of the two connected components of $$X∖{t,s}X\backslash\{t,s\}$$. In fact, the finite number of connected components implies $$C2(t)∪{x}C_2(t)\cup\{x\}$$ is connected, so the same goes for $$(C2(t)∪{x})∪(C2(s)∪{x})(C_2(t)\cup\{x\})\cup(C_2(s)\cup\{x\})$$ : the only thing to verify is the connectedness of $$DD$$. The proof looks like to the first step. If $$U,VU,V$$ are two open sets of $$XX$$ such that $$U∩V∩D=∅U\cap V\cap D=\emptyset$$, $$(U∪V)∩D=D(U\cup V)\cap D=D$$, and $$U∩D≠∅U\cap D\neq\emptyset$$ and $$V∩D≠∅V\cap D\neq\emptyset$$, and if $$u∈U∩Du\in U\cap D$$ and $$v∈V∩Dv\in V\cap D$$, then we show $$X∖{u}X\backslash\{u\}$$ or $$X∖{v}X\backslash\{v\}$$ is not connected.
Finally, if $$XX$$ is a $$33$$-flimsy space and $$x,y,t,sx,y,t,s$$ some distinct points of $$XX$$, then $$DD$$ (defined as previously in $$X∖{y}X\backslash\{y\}$$, a 2-flimsy space) is open and closed in $$X∖{x,t,s}X\backslash\{x,t,s\}$$ and in $$X∖{y,t,s}X\backslash\{y,t,s\}$$, so it is open and closed in $$X∖{t,s}X\backslash\{t,s\}$$, which is not connected. So $$XX$$ is not a 3-flimsy space after all.