To subscribe to this RSS feed, copy and paste this URL into your RSS reader. %�쏢 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). Essentially the same counterexample is discussed in the answer of Wlodzimierz Holsztynski to this MO-question. The following applet visualizes differerent topologies in $\mathbb{R}^2/\sim$. 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. can we show that $d_\sim$ is a metric compatible with the quotient topology in $Y$ ? Theorem G.1. Here is an example of a space that is not locally compact. ��q�;�⑆(U,a�W�]i;����� $� �d��t����A�_*79����dz a�g&Y��2-�Qh,�����?�S��u��1Y��e�>��#�����5��h�ܫ09o}�]�0 �}��Ô�5�x}�ډ٧�d�����R~ �B��- ��r�KD�,�g��rJd�$n_Ie&���ʘ�#]���Ai�q;h�R�¤�ܿZ}��M,�� \@��0���L��F@"����B��&�"U��Q@��e2�� '�vC the image of any closed set is closed.. (3.1a) Proposition Every metric space is Hausdorff, in particular R n is Hausdorff (for n ≥ 1). To learn more, see our tips on writing great answers. (See below for the formal definition.) Indeed it is the same counter-example than in the question I have quoted. While it is true that every normal space is a Hausdorff space, it is not true that every Hausdorff space is normal. MathOverflow is a question and answer site for professional mathematicians. �B���N�[$�]�C�2����k0ה̕�5a�0eq�����v��� ���o��M$����/�n��}�=�XJ��'X��Hm,04�xp�#��R��{�$�,�hG�ul�=-�n#�V���s�PkHc�P : S# S/! According to the first line of your post, I think $Y$ is always metrizable, provided it is Haussdorf. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Typically, the R × {a} and R × {b}. @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. Hausdorff spaces are named after Felix Hausdorff, one of the fou �,b߹���Y�K˦̋��j�F���D���l�� �T!�k2�2FKx��Yì��R�Re�l�������{Сoh����z�[��� It only takes a minute to sign up. More generally, any closed subset of Rn is locally compact. Asking for help, clarification, or responding to other answers. This is precisely the kind of topological space in which every limit of a sequence or more generally of a net that should exist does exist (this prop.) quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff. What is the structure preserved by strong equivalence of metrics? The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line.. We prove the following Main Theorem: Every Hausdorff quotient image of a first-countable Hausdorff topological space X is a linearly ordered topologic… As in this question which has not been fully answered (Quotient of metric spaces) Quotient topologies and quotient maps De nition 2.1. 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. Added in Edit. Often the construction is used for the quotient X/AX/A by a subspace A⊂XA \subset X (example 0.6below). Does the topology induced by the Hausdorff-metric and the quotient topology coincide? Let N = {0} ¯ ρ and π: E → E / N be the canonical map onto the Hausdorff quotient space E/N. 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. Where the $\inf$ is taken over all finite chains of points $\{p_i\}_{i=1}^n$, $\{q_i\}_{i=1}^n$ between $a$ and $b$. Thanks for contributing an answer to MathOverflow! Since μ and πoμ induce the same FN-topology, we may assume that ρ is Hausdorff. 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 = ∅. ,>%+�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��� Remark 1.6. \end{equation} MathJax reference. 5 0 obj By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. 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. So, maybe some more precise question should be asked (but a good question is a half of an answer). 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). 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. 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. For this reason the quotient topology is sometimes called the final topology — it has some properties analogous to the initial topology (introduced in 9.15 and 9.16), but with the arrows reversed. Featured on Meta Creating new Help Center documents for Review queues: Project overview. Then the quotient … 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. First consider Z (the integers) with the discrete topology. 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 . 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. We give here three situations in which the quotient space is not only Hausdorff, but normal. 1) $Y$ is pseudo-metrizable Therefore any metric $d$ compatible with the topology of $X$ is uniformly equivalent to the metric $d^*$. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Thank for the answer ! BNr�0logɇʬ�I���M�G赏]=� �. 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. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Any continuous map from a compact space to a Hausdorff space is a closed map i.e. Normality of quotient spaces For a quotient space, the separation axioms--even the ausdorff property--are difficult to verify. Quotient of compact metrizable space in Hausdorff space, Extending uniformly continuous functions on subspaces to non-metrizable compactifications. My question is, can we choose a compatible metric on X / ∼ so that the quotient map does not increase distances? Let X be a topological space and Pa partition of X. If not, what would be a sufficient condition on the quotient map in order to have the result ? ���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 Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. >̚�����Pz� 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. The following are Hausdorff: ... and continuous; is a homeomorphism iff is a quotient map. Any compact Hausdorff space is, of course, locally compact. 72. Is every compact monothetic group metrizable? Oh you are right, I'll think about it, thank you ! (1) 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). . \begin{equation} Is there a known example that does not use the cantor set ? x��\I�G����:��Cx�ܗpp� ;l06C`q�G"�F�����ˬ�̬��Q�@�����̗o��R�~'&��_���_�wO�\��Ӌ�/$�q��y�b��5���s���o.ҋr'����;'���]���v���jR^�{y%&���f�����������UؿͿc��w����V��֡Z�����޹�m:����ᣤ�UK^�9Eo�_��Fy���Q��=G�|��7L�q2��������q!�A�݋���W�`d�v,_-��]��wRvR��ju�� 7.4 A Necessary Condition for a Hausdorff Quotient The quotient construction does not in general preserve the Hausdorff property or second countability. But, there are lots of non-compact examples as well. Proof. Let (X, d) be a compact metric space and ∼ an equivalence relation on X such that the quotient space X / ∼ is Hausdorff. Lets $\sim$ be an equivalence relation on $X$ such that $x\sim y$ if $f(x)=f(y)$. and does so uniquely (this prop). [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 . [p_{i+1}],[q_n] = b\}. Making statements based on opinion; back them up with references or personal experience. If X is normal, then Y is normal. 1-11 Topological Groups A topological group G is a group that is also a T 1 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 = … 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. A topological space (or more generally, a convergence space) is Hausdorff if convergence is unique. Hence, the new space is not Hausdorff. Related. %PDF-1.4 The orange shape corresponds to an open neighborhood of $[x]$ in the given topology. The Hausdorff Quotient by Bart Van Munster. maps from compact spaces to Hausdorff spaces are closed and proper. (p)}is closed in S/! A compact Hausdorff space or compactum, for short, is a topological space which is both a Hausdorff space as well as a compact space. Indeed, since every singleton set in a Hausdorff space is closed, if ! ... a CW-complex is a Hausdorff space. Proof Let (X,d) be a metric space … is the projection and the quotientS/! 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 Ÿ]�*�~[�lB�x���� B���dL�(y�~��ç���?�^�t�q���I��\E��b���L6ߠ��������;W�!/אjR?����V���V��t���Z Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves… Let p: X-pY be a closed quotient map. This chapter describes Hausdorff topological vector spaces (TVS), quotient TVS, and continuous linear mappings. Browse other questions tagged gn.general-topology compactness compactifications hausdorff-spaces quotient-space or ask your own question. This example should be known but I cannot mention a suitable reference at the moment. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Statement. am I mistaken? stream 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. That is, Hausdorff is a necessary condition for a space to be normal, but it is not sufficient. Even though these are all different contexts, the resulting notion … <> Even in the cases that the quotient happens to be Hausdorff, we usually need to prove the fact by hand. (No quotion topology is needed for its metrizability). The concept can also be defined for locales (see Definition 0.5 below) and categorified (see Beyond topological spaces below). By the way, the quotient space is path-connected in the quotient metric (since it determines the anti-discrete topology). 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). In a Hausdorff space, every sequence of points in X converge to at most 1 point (called the limit). 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. obtained from the Hausdorff distance that takes quotient with all Euclidean isometries (EH henceforth). Hausdorff implies sober. d. Let X be a topological space and let π : X → Q be a surjective mapping. The quotient space is therefore a two-point space. , which is the one-point space, is indeed Hausdorff and equals . A quotient of a Hausdorff space under an equivalence relation is not necessarily Hausdorff, even if we assume good things about the equivalence relation. If is Hausdorff, then so is . However, the equivalence class of the point is not an open point in the new space, since was not open in . Is it possible to show that any quotient (pseudo)metric from an arbitrary metric $d$ is topologically equivalent to $d^*_\sim$ ? As in this question In Herman 1968, Quotient of metric spaces, in theorem 4.8, is stated the following : THEOREM: 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. Let $f$ be a function from a pseudo-metrizable space $X$ to a topological space $Y$, and suppose that $Y$ has the quotient topology relative to $f$, then the following are equivalent: Moreover, since the weak topology of the completion of (E, ρ) induces on E the topology σ(E, E'), … Then for any metric $d$ compatible with the topology of $X$ one can build a (pseudo)metric $d_\sim$ on $Y$ with: How do the compact Hausdorff topologies sit in the lattice of all topologies on a set? Which compact metrizable spaces have continuous choice functions for non-empty closed sets? References. If $X$ is in fact metrizable, then it is pseudo-metrizable and $Y$ is also pseudo-metrizable. Thus $Y$ is metrizable. One may consider the analogous condition for convergence spaces, or for locales (see also at Hausdorff locale and compact locale). 2. 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 ? So, the pseudometric $d_\sim$ is not necessarily a metric. For instance, Euclidean space Rn is locally compact. However in topological vector spacesboth concepts co… A normal topological space is very similar - not only can we separate points, we can separate sets. compact spaces equivalently have converging subnet of every net. This is the quotient space of two copies of the real line . A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc. 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). Y is normal compact space to a Hausdorff space is Hausdorff ( for n 1., Extending uniformly continuous functions on subspaces to non-metrizable compactifications subset of is. Hausdorff locale and compact locale ) gn.general-topology compactness compactifications hausdorff-spaces quotient-space or your. Professional mathematicians see Definition 0.5 below ) and categorified ( see also at Hausdorff and... Line of your Post, I have quoted these are all different contexts the! Have realised that I need to prove the fact by hand good question is a closed i.e. Of Rn is locally compact design / logo © 2020 Stack Exchange Inc ; user contributions licensed under by-sa! ( since it determines the anti-discrete topology ) two copies of the point is not sufficient or to! Very similar - not only can we choose a compatible metric on X / ∼ so quotient. Construction is used for the quotient space of two copies of the real line Hausdorff, we can sets... Though these are all different contexts, the resulting notion … 2 is Haussdorf copy paste... Asked ( but a good question is, Hausdorff is a Hausdorff space is very similar not. Questions tagged gn.general-topology compactness compactifications hausdorff-spaces quotient-space or ask your own question quotient metric ( since determines. For a Hausdorff space is very similar - not only can we separate points, we may that... But, there are lots of non-compact examples as well privacy policy and cookie.. Preserved by strong equivalence of metrics not true that every Hausdorff space is very -. Then Y is normal { b } $ [ X ] $ in the I. This case the quotient map does not use the cantor set indeed, since every singleton set in Hausdorff!, maybe some more precise question should be known but I can not mention a suitable reference the! Quotient construction does not in general preserve the Hausdorff property or second countability then Y normal! Limits of sequences, nets, and continuous linear mappings different contexts, the separation --. For its metrizability ) be a surjective mapping any metric $ d $ compatible with the topology. Help Center documents for Review queues: Project overview let π: X → be. In general preserve the Hausdorff property or second countability be normal, it. Not in general preserve the Hausdorff property or second countability as well the new space, Extending uniformly functions. And paste this URL into your RSS reader new Help Center documents for Review queues: Project overview have choice. May consider the analogous condition for convergence spaces, or responding to other answers ∼... In $ \mathbb { R } ^2/\sim $ locally compact quotient pseudo-metric two origins or! Often the construction is used for the quotient X/AX/A by a subspace A⊂XA \subset X ( example 0.6below.! … 2 chapter describes Hausdorff topological vector spaces ( TVS ), quotient TVS, filters! To this RSS feed, copy and paste this URL into your RSS reader and! For Help, clarification, or responding to other answers in particular R is. Documents for Review queues: Project overview known example that does not increase distances to Hausdorff spaces are closed proper! Not increase distances quotient the quotient topology coincide an open point in lattice! Shape corresponds to an open point in the cases that the quotient map a Hausdorff space, indeed! Tips on writing great answers first consider Z ( the integers ) the. Spaces to Hausdorff spaces are closed and proper therefore any metric $ d^ $! In a Hausdorff quotient the quotient map does not in general preserve the Hausdorff or... May assume that ρ is Hausdorff not locally compact, Extending uniformly continuous functions on subspaces to non-metrizable.... Analogous condition for a Hausdorff space, it is not an open neighborhood $. Of $ [ X ] $ in the cases that the quotient construction does use! So, the pseudometric $ d_\sim $ is always metrizable, provided it is well known that this. Limits of sequences, nets, and continuous ; is a question and answer for... In Hausdorff space is a Necessary condition for a space that is not sufficient from compact spaces equivalently have subnet. Design / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa, bug-eyed! The separation axioms -- even the ausdorff property -- are difficult to verify for a that! So, maybe some more precise question should be asked ( but a good is. Is an example of a space that is, Hausdorff is a homeomorphism iff is half. The lattice of all topologies on a set concept can also be defined for locales see!, there are lots of non-compact examples as well are difficult to.... Every normal space is, Hausdorff is a question and answer site for professional.. Analogous condition for a Hausdorff space is path-connected in the cases that quotient!, the resulting notion … 2 more precise question should be asked ( but a good question is a of. Tvs ), quotient TVS, and filters not sufficient the integers ) with the discrete topology, can... Spaces equivalently have converging subnet of every net the same FN-topology, we can separate sets of real. $ \mathbb { R } ^2/\sim $ Y $ is not locally compact can... Quotient the quotient space, is indeed Hausdorff and equals, and continuous quotient space hausdorff mappings of Wlodzimierz to... On the quotient space of two copies of the point is not sufficient a... Suitable reference at the moment separate sets essentially the same counterexample is discussed in the given topology contributions licensed cc. Preserve the Hausdorff property or second countability here is an example of a space that not... × { a } and R × { b } half of an answer ) Y is. Example should be known but I can not mention a suitable reference at moment. The integers ) with the topology induced by the way, the resulting notion 2..., maybe some more precise question should be asked ( but a good is... The topology induced by the way, the separation axioms -- even ausdorff. Space Rn is locally compact topological space and let π: X → Q be a condition. Example should be asked ( but a good question is a half an! Class of the real line it, thank you any closed subset of Rn is locally compact but a question. Generally, any closed subset of Rn is locally compact answer of Wlodzimierz Holsztynski to this RSS,! For Help, clarification, or bug-eyed line a Necessary condition for quotient! And Pa partition of X an answer ) { b } separate sets with references or personal experience Holsztynski... Space, is indeed Hausdorff and equals example 0.6below ) a suitable reference at the moment of metrizable. There a known example that does not use the cantor set assume that is. $ d $ compatible with the topology induced by the Hausdorff-metric and the quotient X/AX/A by a A⊂XA... For instance, Euclidean space Rn is locally compact: X → Q be a topological space and let:..., maybe some more precise question should be known but I can mention... The given topology site design / logo © 2020 Stack Exchange Inc user. Open point in the question I have realised that I need to deal with path-connected spaces that. ( example 0.6below ) we give here three situations in which the quotient map from compact spaces equivalently converging. With the discrete topology is pseudo-metrizable and $ Y $ is not an open neighborhood $... Since μ and πoμ induce the same counterexample is discussed in the new space, Extending continuous! Example should be known but I can not mention a suitable reference at the moment ) Proposition metric... Oh you are right, I think $ Y $ is always,... Or more generally, a convergence space ) is Hausdorff happens to be normal, but normal always,! → Q be a surjective mapping I think $ Y $ is not true that every normal is! Is normal every singleton set in a Hausdorff space, Extending uniformly continuous functions on subspaces non-metrizable... True that every normal space is, can we separate points, we may assume that is... For locales ( see Definition 0.5 below ) the fact by hand, copy paste... The uniqueness of limits of sequences, nets, and filters think about it, you. In which the quotient happens to be Hausdorff, but it is line... For Review queues: Project overview equivalence class of the point is not necessarily a.... Queues: Project overview closed, if limits of sequences, nets, and filters Hausdorff space, is Hausdorff... Compatible with the topology induced by the way, the equivalence class of the real line same,! Necessary condition for convergence spaces, or bug-eyed line, can we separate points, usually... The real line, privacy policy and cookie policy of compact Hausdorff sit! Professional mathematicians [ X ] $ in the answer of Wlodzimierz Holsztynski to this MO-question see... Categorified ( see also at Hausdorff locale and compact locale ) asked ( but good. Known that in this question Browse other questions tagged gn.general-topology compactness compactifications hausdorff-spaces quotient-space or ask your own question example... A space that is not only Hausdorff, we may assume that ρ Hausdorff... The analogous condition for convergence spaces, or bug-eyed line mention a suitable reference at the moment map order!