A Topological Variation of the Reconstruction Conjecture

Abstract

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible.We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals R, the rationals Q and the irrationals P are reconstructible, as well as spaces occurring as Stone-Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

Reconstructing topological graphs and continua

“…Most importantly, we see that local properties, so topological properties like local compactness or local connectedness, are reconstructible in T 1 spaces, and further, that for compact Hausdorff spaces, reconstructing compactness is equivalent to reconstructing the space itself. Additional information on the topological reconstruction problem can be found in [17].…”

“…For the sphere, stereographic projection gives D(S n ) = {R n }. The second two authors have shown, [17], that all these spaces are reconstructible, as are the rationals, which has deck D(Q) = {Q}, and the irrationals, P , with deck D(P ) = {P }.…”

The space $ω^*$ and the reconstruction of normality

“…The present paper is concerned with the topological version of the reconstruction conjecture, introduced in [21]. A topological space Y is called a card of another space X if Y is homeomorphic to X \ {x} for some x in X.…”

“…It is shown in [21] that all the aforementioned examples are reconstructible, with the exception of the Cantor set where we have D(C) = D(C \ {0}). This example also shows that compactness is a non-reconstructible property.…”

“…The result about reconstruction of normality is somewhat curious, especially so because all other topological separation axioms are easily seen to be reconstructible for spaces containing at least three points [21,Thm. 3.1].…”