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4 rephrase

First of all, the one-point compactification of three open intervals is not a "tack", it's a three-leaf clover. I think that you mean a one-point union of three closed intervals; of course it doesn't matter if the other three endpoints are there or not. This topological type can be called a "Y" or a "T" or a "simple triod". R.L. Moore published a solution to your question in 1928. The answer is no. It was generalized in 1944 by his student Gail Young: You can only have countably many $(n-1)$-dimensional tacks in $\mathbb{R}^n$ for any $n \ge 2$. For her theorem, the name "tack" makes rather more sense, but she calls it a "$T_n$-set".

Actually Moore's theorem applies to a more general kind of triod, in which three tips of the "Y" are connected to the center by "irreducible continua", rather than necessarily intervals.

I don't know whether I might be spoiling a good question, but here in any case is a solution to the original question (see as both Moore and Young did something more general that takes more discussion). Following domotorp's hint, there is a principle of accumulation onto a countable set of outcomes pigeonhole principle for infinite cardinalitiesuncountable sets. If $f:A \to B$ is a function from an uncountable set $A$ to a countable set $B$, then there is an uncountable inverse image $A' = f^{-1}(b)$. If you want to show that $A$ does not exist, then you might as well replace it with $A'$. Unlike the finite pigeonhole principle, which becomes more limited with each such replacement, $A'$ has the same cardinality as $A$, so you haven't lost anything. You are in an equivalent position if you think of a new function $f':A' \even free to B'$apply the uncountable pigeonhole principle again.

Suppose that you have uncountably many simple triods in the plane. Given a simple triod, we can choose a circle $C$ with rational radius and rational center with the branch point of the triod on the inside and the three tips on the outside. Since there are only countably many such circles, there are uncountably many triods with the same circle $C$. We can trim the segments of each such triod so that they stop when they first touch $C$, to make a pie with three slices (a Mercedes-Benz symbol). Then, given such a triod, we can pick a rational point in each of three slices of the pie. Since there are only countably many such triples of points, there must be uncountably many triods with the same three points $p$, $q$, and $r$. In particular there are two such triods, and a suitable version of the Jordan curve theorem implies that they intersect.

Well, the

The argument can be simplified to just pick a rational triangle that functions as the circle, and whose corners function as the three separated points. But I think that there is something to learn from the variations together, namely that the infinite pigeonhole principle gives you a lot of control. For instance, with hardly any creativity, you can assume that the triods are all large.

3 added 124 characters in body

First of all, the one-point compactification of three open intervals is not a "tack", it's a three-leaf clover. I think that you mean a one-point union of three closed intervals; of course it doesn't matter if the other three endpoints are there or not. This topological type can be called a "Y" or a "T" or a "simple triod". R.L. Moore published a solution to your question in 1928. The answer is no. It was generalized in 1944 by his student Gail Young: You can only have countably many $(n-1)$-dimensional tacks in $\mathbb{R}^n$ for any $n \ge 2$. For her theorem, the name "tack" makes rather more sense, but she calls it a "$T_n$-set".

Actually Moore's theorem applies to a more general kind of triod, in which three tips of the "Y" are connected to the center by "irreducible continua", rather than necessarily intervals.

I don't know whether I might be spoiling a good question, but here in any case is a solution to the original question (see as both Moore and Young did something more general that takes more discussion). Following domotorp's hint, there is a "principle of accumulation onto a countable set of outcomes"outcomes pigeonhole principle for infinite cardinalities. Speaking precisely, if If $f:A \to B$ is a function from an uncountable set $A$ to a countable set $B$, then there is an uncountable inverse image $A' = f^{-1}(b)$. If you want to show that $A$ does not exist, then you might as well replace it with $A'$. If you like, you can repeat Unlike the simplification finite pigeonhole principle, which becomes more limited with some each such replacement, you are in an equivalent position if you think of a new function $f':A' \to B'$.

Suppose that you have uncountably many simple triods in the plane. Given a simple triod, we can choose a circle $C$ with rational radius and rational center with the branch point of the triod on the inside and the three tips on the outside. Since there are only countably many such circles, there are uncountably many triods with the same circle $C$. We can trim the segments of each such triod so that they stop when they first touch $C$, to make a pie with three slices (a Mercedes-Benz symbol). Then, given such a triod, we can pick a rational point in each of three slices of the pie. Since there are only countably many such triples of points, there must be uncountably many triods with the same three points $p$, $q$, and $r$. In particular there are two such triods, and a suitable version of the Jordan curve theorem implies that they intersect.

Well, the argument can be simplified to just pick a rational triangle that functions as the circle, and whose corners function as the three separated points. But I think that there is something to learn from the variations together, namely that the infinite pigeonhole principle of accumulation can give gives you a lot of control. For instance, with hardly any creativity, you can assume that the triods are all large.

2 added 1924 characters in body

I don't know whether I might be spoiling a good question, but here in any case is a solution to the original question (see as both Moore and Young did something more general that takes more discussion). Following domotorp's hint, there is a "principle of accumulation onto a countable set of outcomes". Speaking precisely, if $f:A \to B$ is a function from an uncountable set $A$ to a countable set $B$, then there is an uncountable inverse image $A' = f^{-1}(b)$. If you want to show that $A$ does not exist, then you might as well replace it with $A'$. If you like, you can repeat the simplification with some new function $f':A' \to B'$.

Suppose that you have uncountably many simple triods in the plane. Given a simple triod, we can choose a circle $C$ with rational radius and rational center with the branch point of the triod on the inside and the three tips on the outside. Since there are only countably many such circles, there are uncountably many triods with the same circle $C$. We can trim the segments of each such triod so that they stop when they first touch $C$, to make a pie with three slices (a Mercedes-Benz symbol). Then, given such a triod, we can pick a rational point in each of three slices of the pie. Since there are only countably many such triples of points, there must be uncountably many triods with the same three points $p$, $q$, and $r$. In particular there are two such triods, and a suitable version of the Jordan curve theorem implies that they intersect.

Well, the argument can be simplified to just pick a rational triangle that functions as the circle, and whose corners function as the three separated points. But I think that there is something to learn from the variations together, namely that the principle of accumulation can give you a lot of control. For instance, with hardly any creativity, you can assume that the triods are all large.

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