**Question**: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate?

Throughout, $X$ is a continuum. That is to say, a compact connected Hausdorff space. (I'd be interested on any info for non-connected spaces too!)

The story is we want to remove a point $p \in X$ and replace it with a *small* subcontinuum. Formally we want a space $Y$ and continuous map $Y \to X$ such that all fibres are nontrivial save one, and the nontrivial fibre $F$ is a nondegenerate subcontinuum with void interior. The question is trivial if we allow $F$ to have interior as we can always take a copy of $[0,1]$ and glue it to $p$ at an endpoint.

When $X$ is metric this can always be done. For example we can *resolve* $X$ at the point $p$ onto the arc: consider the following subspace of $X \times [-1,1]$.

$\displaystyle Y = \Big \{ \Big (x, \sin \Big (\frac{1}{d(x,p)}\Big ) \Big): x \in X-p \Big\} \bigcup \Big ( \{p\} \times [-1,1]\Big )$.

Then the projection map onto the $x$-coordinate satisfies the above condition.

More generally we can resolve onto any locally connected continuum $F$, by recalling $F$ is the image of an arc and constructing a map from $[0,1)$ that visits each point cofinally many times.

When $X$ is not metric we cannot always do this. For example consider the *long arc* $[0,\omega_1]$. Removing the endpoint $p$ we get the *long line* whose Stone Čech compactification is known to be the same as the one-point compactification. Since the remainder of the Stone Čech maps onto the remainder of every compactification, we cannot replace $p$ with a nondegenerate continuum, as then that continuum would be the image of a point.

It is known exactly when $X-p$ admits a nontrivial remainder. Namely when $X-p$ has two disjoint closed non-compact subsets. Has it been studied whether there are any conditions for when a nontrivial **connected** remainder exists?

I know Smith has given a condition for when the Stone Čech remainder is connected (when $p$ fails to be a local cut point) but this is of no immediate use. For example if $X$ is a circle $(X-p)^*$ has exactly two components but the above shows we can always *wrap* $X-p$ around an arc. In this case we get something like a Warsaw circle.

**Edit:** You might try to generalise the resolution approach to something involving a gauge $\mathcal G $ of pseudometrics on $X$. But this throws up problems if $\rho(p,x)=0$ for some $x \ne p$ and $\rho \in \mathcal G$. The obvious solution is to replace $\mathcal G$ with a gauge that is *positive definite* at $p$ meaning $\rho(p,x)=0 \implies p=x$. But this cannot in general be done. In fact if a single $\rho$ is positive definite then $\{p\}$ is immediately a $G_\delta$ set which is not a guarantee for non-metric spaces.

pseudocompact. $\endgroup$stand-aloneself-contained self-explanatory short formulation of the Question would be helpful. $\endgroup$