We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X . AMS Classification 54F15,54F05,20F65,20E08

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...

❆❜str❛❝t ✕ ❆ss❡ss✐♥❣ s♦❢t✇❛r❡ q✉❛❧✐t2 ❛ttr✐❜✉t❡s ✭s✉❝❤ ❛s ♣❡r❢♦r♠❛♥❝❡✱ r❡❧✐❛❜✐❧✐t2✱ ❛♥❞ s❡❝✉r✐t2✮ ❢r♦♠ s♦✉r❝❡ ❝♦❞❡ ✐s ♦❢ t❤❡ ✉t♠♦st ✐♠♣♦rt❛♥❝❡✳ ❚❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛ s♦❢t✇❛r❡ s2st❡♠ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜2 ✐ts ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥✳ ❚❤❡ ❛✐♠ ♦❢ t❤❡ ♣❛r❛❧❧❡❧ ❛♥❞ ❞✐str✐❜✉t❡❞ ❡1❡❝✉t✐♦♥ ✐s t♦ s♣❡❡❞ ✉♣ ❜2 ♣r♦✈✐❞✐♥❣ t❤❡ ♠❛1✐♠✉♠ ♣♦ss✐❜❧❡ ❝♦♥❝✉rr❡♥❝2 ✐♥ ❡1❡❝✉t✐♥❣ t❤❡ ❞✐str✐❜✉t❡❞ s❡❣♠❡♥ts✳ ■t ✐s ❛...