By John B. Conway

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its light velocity may be priceless to scholars who're nonetheless studying to write down proofs. must haves contain calculus and not less than one semester of study, the place the coed has been safely uncovered to the information of simple set thought resembling subsets, unions, intersections, and features, in addition to convergence and different topological notions within the genuine line. Appendices are integrated to bridge the distance among this new fabric and fabric present in an research path. Metric areas are one of many extra universal topological areas utilized in different components and are for this reason brought within the first bankruptcy and emphasised through the textual content. This additionally conforms to the technique of the ebook to begin with the actual and paintings towards the extra common. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of thought, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact areas.

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**Example text**

2. 1. 2. To see that Σk acts freely on C(k), note that the structure map from S 1 to the cactus deﬁnes an ordering on the lobes. 11, keeping the marked point somewhere on the left to return it to the original position at the end of the move. 12. We will prove that C(k)/Σk is K(RBk , 1) by induction on k. For k = 1 the space C(1)/Σ1 = C(1) is obviously homeomorphic to S 1 × R+ , which is K(Z, 1). Since RB1 ∼ = Z, we get the induction base. , takes a quotient of the cactus by its k+1st lobe. The marked point, if it happened to be on that lobe, will be at the resulting contraction point, assigned to the lobe that follows the one being contracted, if one moves from the marked point along the parameter of the wrapping map from S 1 to the cactus.

This thickening can be thought of as assigning a “width” to the ink used in drawing a fat graph. Thus one is actually drawing a two dimensional space, and it is not hard to see that it is homeomorphic to a smooth surface. 8) of fat graphs which consist of the same underlying graph, but have diﬀerent cyclic orderings at the top vertex. The orderings of the edges are induced by the counterclockwise orientation of the plane. Notice that Γ1 thickens to a surface of genus zero with four boundary components, whereas Γ2 thickens to a surface of genus 1 with two boundary components.

The BV structure arising in string topology at the level of homology comes from an action of a cacti operad C at the motivic level, quite close to the category of topological spaces. The kth component C(k) of the cacti operad C for k ≥ 1 may be described as follows. C(k) is the set of tree-like conﬁgurations of parameterized circles, called the 38 Chapter 2. The cacti operad 2 1 4 3 k 5 ... 10: A cactus lobes, labeled by numbers 1 through k, of varying (positive) radii, along with the following data: (1) the choice of a cyclic order of components at each intersection point and (2) the choice of a marked point on the whole conﬁguration along with the choice of one of the circles on which this point lies.