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I have the following problem:

Check if the following figures are homeomorphic, with the subspace topology of $\mathbb{R}^2.$ enter image description here

I believe that are not homeomorphic, but I can't find an invariant that help me to show it. I tried with the invariant that consist in the number of connected components, but it doesn't work. I hope you can help me. I'm a fool with topology.

Thanks!!!

Antonio Hernandez
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  • Please provide additional context to your question (e.g. what have you done). – WhatsUp Dec 02 '21 at 16:40
  • @EricTowers I saw that, but is that answer correct? – Antonio Hernandez Dec 02 '21 at 16:41
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    Doesn't matter: an incorrect answer there does not justify posting a duplicate question. – Eric Towers Dec 02 '21 at 16:42
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    @AntonioHernandez Yes, the answer there is fine. Of course, there are things assumed known. For example, that cut-points must be mapped to cut-points in a homeomorphism. That the four points where the circles touch in each figure are the only cut-points. That the number of connected components is an invariant. That removing the two cut-points on the ends from the second figure and restricting the homeomorphism gives an homeomorphism with the corresponding image. – plop Dec 02 '21 at 16:50
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    The argument then finishes with the observation that the restricted function is mapping a space with three components to one with four. – plop Dec 02 '21 at 16:53
  • @Boxwood Thank you so much!!!! – Antonio Hernandez Dec 02 '21 at 16:54

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