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I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.

I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove the relative interior of $pq$ from $\mathbb{R}^2$, pull apart the opening and consider the new "boundary" which I will denote by an oriented path $\overrightarrow{pq}$ and $\overrightarrow{qp}$. Along each of $\overrightarrow{pq}$ and $\overrightarrow{qp}$ I want to glue a copy of $\mathbb{R}^2$. The result will be the original $\mathbb{R}^2$ with two additional $\mathbb{R}^2$ glued along $\overrightarrow{pq}$ and $\overrightarrow{qp}$. However, here's the catch, I want $pq$ to act as a "portal" meaning that if I consider a path entering from $\overrightarrow{pq}$ I enter the copy of $\mathbb{R}^2$ that I glued along $\overrightarrow{pq}$ and I can only get back to the original $\mathbb{R}^2$ via that same entrance. Likewise, if I consider a path that enters along $\overrightarrow{qp}$ I should enter the copy of $\mathbb{R}^2$ that I glued along that boundary.

It should be noted that if I can do this construction without removing $pq$ that would be fine as well. I just don't know another way to separate the segment into two distinct parts.

In $\mathbb{R}^2$ I have a smooth orientable patch of surface $\sigma$. (homeomorphic to a closed disc) that I want to behave as a portal. Each side of $\sigma$ should be glued to a different copy of $\mathbb{R}^3$ that is only accessible by a path that intersects $\sigma$ from that side.

I've thought about how to construct such a space with quotient spaces, but the intuition of a quotient space doesn't seem quite right. I've also considered gluing along the limit points of Cauchy sequences that come at $pq$ from a particular side, but that seems difficult to formalize. How would one go about formalizing this type of construction?

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  • $\begingroup$ How about $(\mathbb R^2_{++} \sqcup \mathbb R^2_{+-}) \sqcup (\mathbb R^2_{-+} \sqcup \mathbb R^2_{--})$ modulo $\sim$, where $\sim$ is the equivalence relation generated by $t_{++} \sim t_{+-}$ if $t \not\in (p, q)$ and $t_{+\pm} \sim t_{-\pm}$ if $t \in (p, q)$? $\endgroup$
    – LSpice
    Nov 7, 2014 at 1:47
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    $\begingroup$ This sentence: The result will be the original $\mathbb R^2$ with two additional $\mathbb R^2$ glued along $\overline{pq}$ and $\overline{qp}$--is not clear to me. $\endgroup$ Nov 7, 2014 at 1:48
  • $\begingroup$ @LSpice Admittedly that's a good thought. By equating $\mathbb{R}_{++}^2$ with $\mathbb{R}_{+-}^2$ you're making two copies of $pq$ and then pasting each to a different copy of $pq$. I think this is roughly topologically equivalent to what I want, but it doesn't have the same geometric properties. In particular this notion of orientation on $pq$ such that entering from one side transports you to a different space than entering on the other. $\endgroup$
    – user61430
    Nov 7, 2014 at 2:05
  • $\begingroup$ Notice that I do not equate the two $\mathbb R^2_{+\pm}$. Instead, I glue $t_{++}$ to $t_{+-}$ only if $t \not\in (p, q)$. This "double all, then collapse most" approach has the effect of doubling just $(p, q)$, so that you can tell on which 'side' you entered it. $\endgroup$
    – LSpice
    Nov 7, 2014 at 2:07
  • $\begingroup$ @WłodzimierzHolsztyński I want to remove the relative interior of $pq$ and consider the oriented paths $\overrightarrow{pq}$ and $\overrightarrow{qp}$. Along each path I want to glue a copy of $\mathbb{R}^2$. Thus the result is three copies of $\mathbb{R}^2$ glued along $pq$. $\endgroup$
    – user61430
    Nov 7, 2014 at 2:07

1 Answer 1

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Based on your comment at Creating topological spaces with portals, I think that I may have an answer. However, it seems to me that you are asking me to capture the right intuitive generalisation of a concept that you have given only for $\mathbb R^2$; so, if it doesn't fit what you want, then I hope you'll give at least another example.

I think that the general idea is that you want to choose a subset $P$ of a topological space $X$, and glue 3 copies of $X$ along $P$ in such a way that entering $P$ at its 'left' side on the 'original' $X$ goes into the 'left new' $X$, and similarly on the right.

To that end, choose two subsets $H_+$ and $H_-$ of $X$ (probably you want them in $\operatorname{cl}(\complement P)$) such that $H_+ \cup H_- \cup P = X$. (As with the overlap at $p$ and $q$ in your example, we do not require the various sets to be disjoint.) Then "$X$ with portal $P$ via $H_+$ and $H_-$" is defined to be $(H_+ \sqcup H_- \sqcup P) \sqcup (X_+ \sqcup X_-)$, modulo glueing $H_+$, $H_-$, and $P$ along their pairwise intersections, $H_\pm$ pointwise to its copy in $X_\pm$, and $P$ to its copy in each $X_\pm$.

EDIT: Corrected the glueing.

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  • $\begingroup$ Can you clarify the notation $CP$ in $\operatorname{cl}CP$? $\endgroup$
    – user61430
    Nov 7, 2014 at 2:42
  • $\begingroup$ @user61430, it's \complement. $\endgroup$
    – LSpice
    Nov 7, 2014 at 2:43
  • $\begingroup$ Yes, I do believe this answers my question. Thank you! $\endgroup$
    – user61430
    Nov 7, 2014 at 2:59

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