Quotient Spaces and Quotient Maps
... homeomorphism; functional composition is associative; the identity map I : X → X is the identity element in the group of homeomorphisms.) The group of all self-homeomorphisms of X may have interesting subgroups. When we specify some [sub]group of homeomorphisms of X that is isomorphic to some abstra ...
... homeomorphism; functional composition is associative; the identity map I : X → X is the identity element in the group of homeomorphisms.) The group of all self-homeomorphisms of X may have interesting subgroups. When we specify some [sub]group of homeomorphisms of X that is isomorphic to some abstra ...
Free full version - topo.auburn.edu
... and f |S n − {p} is a local homeomorphism. Must f be a homeomorphism? In considering this question, Lelek and Mycielski [3] gave the following theorems: Theorem 2 [3]. If (1) X is connected and X or Y is locally connected, (2) f : X → Y is an open local homeomorphism onto Y , (3) every point p ∈ Y i ...
... and f |S n − {p} is a local homeomorphism. Must f be a homeomorphism? In considering this question, Lelek and Mycielski [3] gave the following theorems: Theorem 2 [3]. If (1) X is connected and X or Y is locally connected, (2) f : X → Y is an open local homeomorphism onto Y , (3) every point p ∈ Y i ...
CLASS NOTES MATH 527 (SPRING 2011) WEEK 3 1. Mon, Jan. 31
... weak Hausdorff space is compactly generated if a subset C ⊆ X is closed if (and only if) for every continuous map g : K −→ X with K compact, the subset g −1 (C) is closed in K. Any time from now on that we talk about spaces, we really mean compactly generated weak Hausdorff spaces. There are a coupl ...
... weak Hausdorff space is compactly generated if a subset C ⊆ X is closed if (and only if) for every continuous map g : K −→ X with K compact, the subset g −1 (C) is closed in K. Any time from now on that we talk about spaces, we really mean compactly generated weak Hausdorff spaces. There are a coupl ...
1 Weak Topologies
... 3. The collection {fα | α ∈ A} separates points from closed sets in X if whenever B is closed in X and x 6∈ B there exists α ∈ A such that fα (x) 6∈ fα (B). Recall that the continuous functions from a completely regular space X to [0, 1] separate points from closed sets, and Urysohn’s theorem shows ...
... 3. The collection {fα | α ∈ A} separates points from closed sets in X if whenever B is closed in X and x 6∈ B there exists α ∈ A such that fα (x) 6∈ fα (B). Recall that the continuous functions from a completely regular space X to [0, 1] separate points from closed sets, and Urysohn’s theorem shows ...
![arXiv:math/9811003v1 [math.GN] 1 Nov 1998](http://s1.studyres.com/store/data/000335093_1-58b56f331c623a9d53167347869d2e0a-300x300.png)
arXiv:math/9811003v1 [math.GN] 1 Nov 1998
... Example. The usual space of Reals, (R, µ) is rim-scattered but not N-scattered. Certainly, the usual base of bounded open intervals has the property that nonempty boundaries of its members are scattered. However, the nowhere dense Cantor set is dense-in-itself. Another example of a rim-scattered sp ...
... Example. The usual space of Reals, (R, µ) is rim-scattered but not N-scattered. Certainly, the usual base of bounded open intervals has the property that nonempty boundaries of its members are scattered. However, the nowhere dense Cantor set is dense-in-itself. Another example of a rim-scattered sp ...
Chapter Three
... by noting that the exponential map E → S 1 defined earlier is in fact a quotient map with the sets E −1 (z), z ∈ S 1 being the orbits of this group action. Example 3.12. We can get a similar action by letting Zn act on Rn by n · x = x + n. The quotient space is the n-torus (S 1 i)n . Example 3.13. E ...
... by noting that the exponential map E → S 1 defined earlier is in fact a quotient map with the sets E −1 (z), z ∈ S 1 being the orbits of this group action. Example 3.12. We can get a similar action by letting Zn act on Rn by n · x = x + n. The quotient space is the n-torus (S 1 i)n . Example 3.13. E ...
