quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff. ��q�;�⑆(U,a�W�]i;����� $� �d��t����A�_*79����dz a�g&Y��2-�Qh,�����?�S��u��1Y��e�>��#�����5��h�ܫ09o}�]�0 �}��Ô�5�x}�ډ٧�d�����R~ Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y → Z is any function, then f is continuous if and only if f ∘ q is continuous. Let (X, d) be a compact metric space and ∼ an equivalence relation on X such that the quotient space X / ∼ is Hausdorff. From uniform equivalent metrics, maybe there is a relation between their corresponding quotient pseudo-metrics but I am stucked here, do you have any idea/theorem/reference that would help me ? First consider Z (the integers) with the discrete topology. However in topological vector spacesboth concepts co… Proof Let (X,d) be a metric space … Roughly, the EH distance attempts to find the optimal Euclidean isometry that aligns the two shapes (in Euclidean space) under the Hausdorff distance.1 We prove important and interesting results about this connection. <> is Hausdorff, then for any p" S,itsimage{! Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves… Making statements based on opinion; back them up with references or personal experience. A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc. My question is, can we choose a compatible metric on X / ∼ so that the quotient map does not increase distances? Lets $\sim$ be an equivalence relation on $X$ such that $x\sim y$ if $f(x)=f(y)$. obtained from the Hausdorff distance that takes quotient with all Euclidean isometries (EH henceforth). %�쏢 (3.1a) Proposition Every metric space is Hausdorff, in particular R n is Hausdorff (for n ≥ 1). Since μ and πoμ induce the same FN-topology, we may assume that ρ is Hausdorff. A topological space (or more generally, a convergence space) is Hausdorff if convergence is unique. Use MathJax to format equations. References. 2) There exists a pseudo-metric $\rho$ compatible with the topology in $X$ such that the quotient pseudo-metric $\rho_\sim$, defined as in (1), is in fact compatible with the quotient topology of $Y$ (The definition of the quotient pseudo-metric by Herman should be equivalent to the one introduced earlier). d. Let X be a topological space and let π : X → Q be a surjective mapping. Definition A topological space X is Hausdorff if for any x,y ∈ X with x 6= y there exist open sets U containing x and V containing y such that U T V = ∅. As in this question , which is the one-point space, is indeed Hausdorff and equals . However, the equivalence class of the point is not an open point in the new space, since was not open in . A topological space X is said to be Hausdorff if, given any two distinct points x and y of X, there is a neighborhood U of x and a neighborhood V of y which do not intersect—for example, U ∩V = ø. We give here three situations in which the quotient space is not only Hausdorff, but normal. The Hausdorff Quotient by Bart Van Munster. Does the topology induced by the Hausdorff-metric and the quotient topology coincide? In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry , in particular as the Zariski topology on an algebraic variety or the spectrum of a ring . Where the $\inf$ is taken over all finite chains of points $\{p_i\}_{i=1}^n$, $\{q_i\}_{i=1}^n$ between $a$ and $b$. But, there are lots of non-compact examples as well. Thank for the answer ! \begin{equation} Beware that quotient objects in the category Vect of vector spaces also traditionally called ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. compact spaces equivalently have converging subnet of every net. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. Applications. am I mistaken? It is well known that in this case the quotient is metrizable. Thanks for contributing an answer to MathOverflow! Here is an example of a space that is not locally compact. Any compact Hausdorff space is, of course, locally compact. Any surjective continuous map from a compact space to a Hausdorff space is a quotient map; Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding This chapter describes Hausdorff topological vector spaces (TVS), quotient TVS, and continuous linear mappings. \end{equation} (p)}is closed in S/! Asking for help, clarification, or responding to other answers. What is the structure preserved by strong equivalence of metrics? d_\sim(a,b) = \inf\{d(p_1,q_1) + \cdots+ d(p_n,q_n);[p_1] = a,[q_i] = Is there a known example that does not use the cantor set ? A Hausdorff space is often called T2, since this condition came second in the original list of four separation axioms (there are more now) satisfied by metric spaces. In topologyand related areas of mathematics, the quotient spaceof a topological spaceunder a given equivalence relationis a new topological space constructed by endowing the quotient setof the original topological space with the quotient topology, that is, with the finest topologythat makes continuousthe canonical projection map(the function that maps points to their equivalence classes). However, I have realised that I need to deal with path-connected spaces so that quotient space is path-connected in the quotient pseudo-metric. Statement. To learn more, see our tips on writing great answers. MathOverflow is a question and answer site for professional mathematicians. Hence, the new space is not Hausdorff. While it is true that every normal space is a Hausdorff space, it is not true that every Hausdorff space is normal. �B���N�[$�]�C�2����k0ה̕�5a�0eq�����v��� ���o��M$����/�n��}�=�XJ��'X��Hm,04�xp�#��R��{�$�,�hG�ul�=-�n#�V���s�PkHc�P According to the first line of your post, I think $Y$ is always metrizable, provided it is Haussdorf. Is every compact monothetic group metrizable? Indeed it is the same counter-example than in the question I have quoted. ���w��#c��V�� -Rr��o�i#���! Thus $Y$ is metrizable. So, maybe some more precise question should be asked (but a good question is a half of an answer). The following applet visualizes differerent topologies in $\mathbb{R}^2/\sim$. Ÿ]�*�~[�lB�x���� B���dL�(y�~��ç���?�^�t�q���I��\E��b���L6ߠ��������;W�!/אjR?����V���V��t���Z @VMrcel You can extend the Cantor starcase function to a continuous function on the closed interval and then you will get a continuous function between closed intervals, for which the quotient pseudometric still is zero. 5 0 obj Let N = {0} ¯ ρ and π: E → E / N be the canonical map onto the Hausdorff quotient space E/N. Therefore, from the theorem there exists a pseudo-metric $d^*$ compatible with the topology in $X$ (it is a metric as $X$ is Hausdorff) such that the quotient pseudo-metric $d^*_\sim$ is compatible with the topology in $Y$ (it is also a metric because $Y$ is Hausdorff too). The quotient space is therefore a two-point space. rev 2020.12.10.38158, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Quotient topologies and quotient maps De nition 2.1. Moreover, since the weak topology of the completion of (E, ρ) induces on E the topology σ(E, E'), … To subscribe to this RSS feed, copy and paste this URL into your RSS reader. [6] In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry , in particular as the Zariski topology on an algebraic variety or the spectrum of a ring . is the projection and the quotientS/! Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. So, the pseudometric $d_\sim$ is not necessarily a metric. We prove the following Main Theorem: Every Hausdorff quotient image of a first-countable Hausdorff topological space X is a linearly ordered topologic… Even though these are all different contexts, the resulting notion … BNr�0logɇʬ�I���M�G赏]=� �. 72. If X is normal, then Y is normal. A compact Hausdorff space or compactum, for short, is a topological space which is both a Hausdorff space as well as a compact space. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. %PDF-1.4 Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. One may consider the analogous condition for convergence spaces, or for locales (see also at Hausdorff locale and compact locale). By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Let X be a topological space and Pa partition of X. . Point Set Topology: Let X be the real line and consider the equivalence relation: xRy iff x and y differ by a multiple of 2^k (k an integer). MathJax reference. can we show that $d_\sim$ is a metric compatible with the quotient topology in $Y$ ? Points x,x0 ∈ X lie in the same G-orbit if and only if x0 = x.g for some g ∈ G. Indeed, suppose x and x0 lie in the G-orbit of a point x 0 ∈ X, so x = x 0.γ and x0 = … If $X$ is in fact metrizable, then it is pseudo-metrizable and $Y$ is also pseudo-metrizable. : S# S/! (1) the image of any closed set is closed.. We know that $X$ is metrizable and compact, thus there is a unique uniform structure in it and all metrics compatible with the topology are uniformly equivalent. ,>%+�wIz� ܦ�p��OYJ��t (����~.�ۜ�q�mvW���6-�Y�����'�լ%/��������%�K��k�X�cp�Z��D�y5���=�ׇ߳��,���{�aj�b����(I�{ ��Oy�"(=�^����.ե��j�·8�~&�L�vյR��&�-fgmm!ee5���C�֮��罓B�Y��� The orange shape corresponds to an open neighborhood of $[x]$ in the given topology. maps from compact spaces to Hausdorff spaces are closed and proper. ���Q���b������%����(z�M�2λ�D��7�M�z��'��+a�����d���5)m��>�'?�l����Eӎ�;���92���=��u� � I����շS%B�=���tJ�xl�����`��gZK�PfƐF3;+�K For the Cantor starcase function $f:C\to[0,1]$ from the standard ternary Cantor set $C$ onto the interval $[0,1]$ and for the standard Euclidean metric $d$ on $C$ the quotient pseudometric $d_\sim$ is constant zero (this follows from the fact that the Cantor set $C$ has length zero). In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Often the construction is used for the quotient X/AX/A by a subspace A⊂XA \subset X (example 0.6below). It only takes a minute to sign up. As in this question which has not been fully answered (Quotient of metric spaces) >̚�����Pz� quotient X/G is the set of G-orbits, and the map π : X → X/G sending x ∈ X to its G-orbit is the quotient map. The following are Hausdorff: ... and continuous; is a homeomorphism iff is a quotient map. Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. ... a CW-complex is a Hausdorff space. 1) $Y$ is pseudo-metrizable Theorem G.1. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Browse other questions tagged gn.general-topology compactness compactifications hausdorff-spaces quotient-space or ask your own question. Related. Any continuous map from a compact space to a Hausdorff space is a closed map i.e. The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line.. Let p: X-pY be a closed quotient map. The concept can also be defined for locales (see Definition 0.5 below) and categorified (see Beyond topological spaces below). For instance, Euclidean space Rn is locally compact. Proof. (See below for the formal definition.) Examples Line with two origins. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … If not, what would be a sufficient condition on the quotient map in order to have the result ? The quotient topology on Pis the collection T= fOˆPj[Ois open in Xg: Thus the open sets in the quotient topology are collections of subsets whose union is open in X. Quotient of compact metrizable space in Hausdorff space, Extending uniformly continuous functions on subspaces to non-metrizable compactifications. If is Hausdorff, then so is . Hausdorff spaces are named after Felix Hausdorff, one of the fou Added in Edit. More generally, any closed subset of Rn is locally compact. An open point in the answer of Wlodzimierz Holsztynski to this RSS,! Singleton set in a Hausdorff space, it is Haussdorf is needed for its metrizability.! Agree to our terms of service, privacy policy and cookie policy visualizes differerent topologies in $ {. Our tips on writing great answers:... and continuous linear mappings a topological space ( more!, you agree to our terms of service, privacy policy and policy. Normal, then for any p '' S, itsimage { Hausdorff is a half of an ). 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