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Let the unit arc be,

$$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$

There is something I found curious about the unit arc which is that,

  • It has an empty interior viewed as a subset of $\mathbb{R}^2$

  • It has an empty relative interior viewed as a subset of its affine hull, which is again a subset of $\mathbb{R}^2$ and inherits the open balls.

But intuitively, the arc is not empty (everything in between the arc can be considered as an interior). Is there any way to define interior for these objects? (unit arc, embedded sphere in the orthant, etc)

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  • $\begingroup$ Its inherited subspace topology. $\endgroup$
    – efs
    Commented Oct 13, 2019 at 4:59
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    $\begingroup$ Just use the relative topo... wait $\endgroup$
    – Ville Salo
    Commented Oct 13, 2019 at 6:33
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    $\begingroup$ As a 1 dimensional manifold. $\endgroup$
    – user43326
    Commented Oct 13, 2019 at 6:47
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    $\begingroup$ Can some one Who understand the question explain what the question is about? I fail to understand the question.. $\endgroup$
    – Praphulla Koushik
    Commented Oct 13, 2019 at 13:58
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    $\begingroup$ @EFinat-S The issue with that is that everything counts as interior in that case. The OP wants to distinguish between the boundary points and the rest. Per Bjorn's answer, this can be done by paying attention to the (lack of) local Euclidean-ness. That said, I don't see how this question is appropriate for MO - it seems a much better fit for MSE. $\endgroup$
    – Noah Schweber
    Commented Oct 13, 2019 at 17:58

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This seems well described by the notion of interior $\mathrm{Int\ }M$ of a manifold with boundary $M$.

See the Wikipedia subentry Manifold: Boundary and interior.

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