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Theorem. The Stone-Čech compactification of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting about this method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.

Theorem. The Stone-Čech compactification of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two parts:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The assumptions $\neg$CH and CH are critical to the constructions. Also the types of continua constructed are very different in Case 1 vs Case 2. In fact, all continua of the type constructed for Case 1 are homeomorphic under CH.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting is that both cases are necessary to this proof because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.

Theorem. The Stone-Čech remainder of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting about this method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.

Theorem. The Stone-Čech compactification of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting about this method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.

Theorem. The Stone-Čech remainder of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c=\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting about this proof method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This is possibly different from the use of RH and $\neg$RH or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then the other half of your proof was superfluous.

Theorem. The Stone-Čech remainder of the real line contains $2^\mathfrak c$ topologically distinct continua.

Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.

The proof was divided into two cases:

Case 1: The Continuum Hypothesis fails ($\mathfrak c>\omega_1$).

Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).

Case 2: The Continuum Hypothesis holds ($\mathfrak c\leq\omega_1$).

Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).

The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.

This is the only theorem I know of which was proved using CH in this way. What's especially interesting about this method is that we know both cases are necessary because CH and $\neg$CH are each consistent with ZFC. This may be different from the use of RH and $\neg$RH, or some other conjecture and its negation. If the conjecture is eventually proved, for instance, then you could throw away the other half of your proof.