Misplaced Pages

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61-619: Valerie Aurora , already an activist for women in open source, joined Mary Gardiner and members of Geek Feminism to develop anti-harassment policies for conferences after Noirin Shirley was sexually assaulted at ApacheCon 2010. Aurora quit her job as a Linux kernel developer at Red Hat and, with Gardiner, founded the Ada Initiative in February 2011. In 2014, Valerie Aurora announced her intent to step down as executive director of

122-530: A 2 , b 2 } , max { a 2 , b 2 } ) , {\displaystyle f(U)=(\min\{a^{2},b^{2}\},\max\{a^{2},b^{2}\}),} where this open interval is an open subset of both R {\displaystyle \mathbb {R} } and Im ⁡ f := f ( R ) = [ 0 , ∞ ) . {\displaystyle \operatorname {Im} f:=f(\mathbb {R} )=[0,\infty ).} However, if U = (

183-535: A , b ) {\displaystyle U=(a,b)} is any open interval in R {\displaystyle \mathbb {R} } that contains 0 {\displaystyle 0} then f ( U ) = [ 0 , max { a 2 , b 2 } ) , {\displaystyle f(U)=[0,\max\{a^{2},b^{2}\}),} which is not an open subset of f {\displaystyle f} 's codomain R {\displaystyle \mathbb {R} } but

244-495: A compact space X {\displaystyle X} to a Hausdorff space Y {\displaystyle Y} is closed and proper (meaning that preimages of compact sets are compact). A variant of the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper then it is also closed. In complex analysis , the identically named open mapping theorem states that every non-constant holomorphic function defined on

305-575: A connected open subset of the complex plane is an open map. The invariance of domain theorem states that a continuous and locally injective function between two n {\displaystyle n} -dimensional topological manifolds must be open. Invariance of domain  —  If U {\displaystyle U} is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}}

366-468: A file systems developer as well as a part-time science writer and Linux consultant. Already an activist for women in open source, she joined Mary Gardiner and members of the Geek Feminism blog to develop anti-harassment policies for conferences after Noirin Shirley was sexually assaulted at ApacheCon 2010. Aurora quit her job as a Linux kernel developer at Red Hat and, with Gardiner, founded

427-709: A function between topological spaces . If S {\displaystyle S} is any set then f ( S ) := { f ( s )   :   s ∈ S ∩ domain ⁡ f } {\displaystyle f(S):=\left\{f(s)~:~s\in S\cap \operatorname {domain} f\right\}} is called the image of S {\displaystyle S} under f . {\displaystyle f.} There are two different competing, but closely related, definitions of " open map " that are widely used, where both of these definitions can be summarized as: "it

488-594: A map. Given any subset T ⊆ Y , {\displaystyle T\subseteq Y,} if f : X → Y {\displaystyle f:X\to Y} is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, surjective ) map then the same is true of its restriction f | f − 1 ( T )   :   f − 1 ( T ) → T {\displaystyle f{\big \vert }_{f^{-1}(T)}~:~f^{-1}(T)\to T} to

549-404: A relatively open map is a strongly open map if and only if its image is an open subset of its codomain. In summary, By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition. The discussion above will also apply to closed maps if each instance of the word "open" is replaced with

610-550: Is an open subset of Im ⁡ f = [ 0 , ∞ ) . {\displaystyle \operatorname {Im} f=[0,\infty ).} Because the set of all open intervals in R {\displaystyle \mathbb {R} } is a basis for the Euclidean topology on R , {\displaystyle \mathbb {R} ,} this shows that f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} }

671-459: Is a point of openness for f , {\displaystyle f,} which by definition means that for every open neighborhood U {\displaystyle U} of x , {\displaystyle x,} f ( U ) {\displaystyle f(U)} is a neighborhood of f ( x ) {\displaystyle f(x)} in Y {\displaystyle Y} (note that

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732-403: Is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition. If f : X → Y {\displaystyle f:X\to Y} is a continuous map that is also open or closed then: In the first two cases, being open or closed is merely a sufficient condition for the conclusion that follows. In the third case, it

793-477: Is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed. The composition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map. However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed. If f : X → Y {\displaystyle f:X\to Y}

854-405: Is a map that sends open sets to open sets." The following terminology is sometimes used to distinguish between the two definitions. A map f : X → Y {\displaystyle f:X\to Y} is called a Every strongly open map is a relatively open map. However, these definitions are not equivalent in general. A surjective map is relatively open if and only if it

915-413: Is a strongly closed map. If in the open set definition of " continuous map " (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") is equivalent to continuity. This does not happen with the definition of "open map" (which is: "every image of an open set

976-499: Is a subset of a topological space then let S ¯ {\displaystyle {\overline {S}}} and Cl ⁡ S {\displaystyle \operatorname {Cl} S} (resp. Int ⁡ S {\displaystyle \operatorname {Int} S} ) denote the closure (resp. interior ) of S {\displaystyle S} in that space. Let f : X → Y {\displaystyle f:X\to Y} be

1037-407: Is always an open subset of X , {\displaystyle X,} the image f ( X ) = Im ⁡ f {\displaystyle f(X)=\operatorname {Im} f} of a strongly open map f : X → Y {\displaystyle f:X\to Y} must be an open subset of its codomain Y . {\displaystyle Y.} In fact,

1098-416: Is an injective continuous map , then V := f ( U ) {\displaystyle V:=f(U)} is open in R n {\displaystyle \mathbb {R} ^{n}} and f {\displaystyle f} is a homeomorphism between U {\displaystyle U} and V . {\displaystyle V.} In functional analysis ,

1159-440: Is an almost open map then it will be an open map if it satisfies the following condition (a condition that does not depend in any way on Y {\displaystyle Y} 's topology σ {\displaystyle \sigma } ): If the map is continuous then the above condition is also necessary for the map to be open. That is, if f : X → Y {\displaystyle f:X\to Y}

1220-409: Is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying the codomain is essential. Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map

1281-476: Is called a relatively closed map if whenever C {\displaystyle C} is a closed subset of the domain X {\displaystyle X} then f ( C ) {\displaystyle f(C)} is a closed subset of f {\displaystyle f} 's image Im ⁡ f := f ( X ) , {\displaystyle \operatorname {Im} f:=f(X),} where as usual, this set

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1342-409: Is closed in R 2 , {\displaystyle \mathbb {R} ^{2},} but p 1 ( A ) = R ∖ { 0 } {\displaystyle p_{1}(A)=\mathbb {R} \setminus \{0\}} is not closed in R . {\displaystyle \mathbb {R} .} However, for a compact space Y , {\displaystyle Y,}

1403-501: Is closed. The inverse of a bijective continuous map is a bijective open/closed map (and vice versa). A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. All local homeomorphisms , including all coordinate charts on manifolds and all covering maps , are open maps. Closed map lemma  —  Every continuous function f : X → Y {\displaystyle f:X\to Y} from

1464-457: Is continuous if the preimage of every open set of Y {\displaystyle Y} is open in X . {\displaystyle X.} (Equivalently, if the preimage of every closed set of Y {\displaystyle Y} is closed in X {\displaystyle X} ). Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn . If S {\displaystyle S}

1525-436: Is continuous, closed, and relatively open, but not (strongly) open. This is because if U = ( a , b ) {\displaystyle U=(a,b)} is any open interval in f {\displaystyle f} 's domain R {\displaystyle \mathbb {R} } that does not contain 0 {\displaystyle 0} then f ( U ) = ( min {

1586-407: Is endowed with the subspace topology induced on it by f {\displaystyle f} 's codomain Y . {\displaystyle Y.} A map f : X → Y {\displaystyle f:X\to Y} is called a closed map or a strongly closed map if it satisfies any of the following equivalent conditions: A surjective map

1647-409: Is freely available, although the Ada Initiative also offered facilitators to conduct the workshops in person. By encouraging women's participation in open-source culture, the Ada Initiative encouraged women to engage in open source professionally and full-time- not just as volunteers. The organization also researched women's roles and experiences in open source, focusing on bringing research up to date;

1708-556: Is guaranteed in general, whereas for preimages, equality f − 1 ( Y ∖ S ) = f − 1 ( Y ) ∖ f − 1 ( S ) {\displaystyle f^{-1}(Y\setminus S)=f^{-1}(Y)\setminus f^{-1}(S)} always holds. The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = x 2 {\displaystyle f(x)=x^{2}}

1769-428: Is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : X → Y {\displaystyle f:X\to Y}

1830-481: Is open and closed, but not continuous. This example shows that the image of a connected space under an open or closed map need not be connected. Whenever we have a product of topological spaces X = ∏ X i , {\textstyle X=\prod X_{i},} the natural projections p i : X → X i {\displaystyle p_{i}:X\to X_{i}} are open (as well as continuous). Since

1891-539: Is open if for any open set U {\displaystyle U} in X , {\displaystyle X,} the image f ( U ) {\displaystyle f(U)} is open in Y . {\displaystyle Y.} Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa. Open and closed maps are not necessarily continuous . Further, continuity

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1952-609: Is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in general not equivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any set S , {\displaystyle S,} only f ( X ∖ S ) ⊇ f ( X ) ∖ f ( S ) {\displaystyle f(X\setminus S)\supseteq f(X)\setminus f(S)}

2013-488: Is relatively open but not (strongly) open. If Y {\displaystyle Y} has the discrete topology (that is, all subsets are open and closed) then every function f : X → Y {\displaystyle f:X\to Y} is both open and closed (but not necessarily continuous). For example, the floor function from R {\displaystyle \mathbb {R} } to Z {\displaystyle \mathbb {Z} }

2074-405: Is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent. By definition, the map f : X → Y {\displaystyle f:X\to Y} is a relatively closed map if and only if the surjection f : X → Im ⁡ f {\displaystyle f:X\to \operatorname {Im} f}

2135-443: Is strongly open (respectively, strongly closed) and g : Y → Z {\displaystyle g:Y\to Z} is relatively open (respectively, relatively closed) then g ∘ f : X → Z {\displaystyle g\circ f:X\to Z} is relatively open (respectively, relatively closed). Let f : X → Y {\displaystyle f:X\to Y} be

2196-411: Is strongly open; so for this important special case the definitions are equivalent. More generally, a map f : X → Y {\displaystyle f:X\to Y} is relatively open if and only if the surjection f : X → f ( X ) {\displaystyle f:X\to f(X)} is a strongly open map. Because X {\displaystyle X}

2257-423: The f {\displaystyle f} -saturated subset f − 1 ( T ) . {\displaystyle f^{-1}(T).} The categorical sum of two open maps is open, or of two closed maps is closed. The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed. A bijective map is open if and only if it

2318-578: The Ada Initiative in February 2011. The organization was named after Ada Lovelace , who worked with Charles Babbage and is considered to be the world's first computer programmer. Two years later, Aurora founded Double Union , a hackerspace for women, with Amelia Greenhall and Liz Henry , but was banned in 2018. The Ada Initiative was shut down in October 2015. Maintaining a blog since 2007, Aurora has written extensively about coding and

2379-461: The Union file system . Her birth name was Val Henson, but she changed it shortly before 2009, choosing her middle name after the computer scientist Anita Borg . In 2012, Aurora, and Ada Initiative co-founder Mary Gardiner were named two of the most influential people in computer security by SC Magazine . In 2013, she won the O'Reilly Open Source Award . Daughter of Carolyn Meinel , Aurora

2440-487: The ext2 and ext3 Linux file systems. While working at Intel , she implemented the ext2 dirty bit and relative atime . Along with Arjan van de Ven, she came up with the idea for ChunkFS, which simplifies file system checks by dividing the file system into independent pieces. She also co-organized the first Linux File Systems Workshop in order to figure out how to spread awareness of and raise funding for file system development. As of 2009, she worked for Red Hat as

2501-668: The open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map. This theorem has been generalized to topological vector spaces beyond just Banach spaces. A surjective map f : X → Y {\displaystyle f:X\to Y} is called an almost open map if for every y ∈ Y {\displaystyle y\in Y} there exists some x ∈ f − 1 ( y ) {\displaystyle x\in f^{-1}(y)} such that x {\displaystyle x}

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2562-528: The Ada Initiative and Blue. Valerie Aurora Valerie Anita Aurora is an American software engineer and feminist activist. She was the co-founder of the Ada Initiative , a non-profit organization that sought to increase women's participation in the free-culture movement , open-source technology , and open source culture. Aurora is also known within the Linux community for advocating new developments in filesystems in Linux, including ChunkFS and

2623-491: The Ada Initiative announced that the organization would close in mid-October, 2015. The announcement described the leadership challenge facing the Initiative: neither co-founder intended to continue as executive director. According to the post on the Ada Initiative website: We felt the likelihood of finding a new ED who could effectively fit into Valerie’s shoes was low. We also considered several other options for continuing

2684-400: The Ada Initiative developed anti-harassment policies for conferences. The Ada Initiative also worked with open source conference organizers to adopt, create and communicate policies to make conferences safer and more inviting for all attendees, particularly women. Conferences such as Ubuntu Developer Summits and all Linux Foundation events, including LinuxCon , have adopted policies based on

2745-459: The Ada Initiative's consideration of the subject as off-topic for a security conference. The abrupt cancellation provoked intense discussion in the information security industry. Since the event at B-Sides SF, lead organizer Ian Fung has outlined his account of the interactions between Blue, Aurora, and the Ada Initiative on the B-Sides SF front page, contradicting some of the claims made by both

2806-611: The Ada Initiative's work. The Ada Initiative developed policy framework for creating a Women in Open Source Scholarship and programming guides for outreach projects and events. The organization also hosted workshops and training. These workshops and programs consisted of Allies Workshops for male and institutional supporters and "First Patch Week" programs, which encourages women's participation in Free and open source software (FOSS) through mentoring. The workshop framework

2867-546: The Ada Initiative, and an executive search committee was formed to find her replacement. Mary Gardiner, deputy executive director, chose not to be a candidate. The committee, headed by Sumana Harihareswara and Mary Gardiner, announced in March 2015 that the Ada Initiative had hired Crystal Huff as the new executive director. Huff, formerly of Luminoso in Boston , continued to work from Massachusetts in her new role. In August 2015,

2928-525: The experiences of women in open source. This has included descriptions of DEF CON and the harassment that took place there. In 2013, Aurora provided a comment to The Verge about the Electronic Frontier Foundation's involvement in the legal defense of Andrew Auernheimer , who was in prison for hacking and had previously harassed Kathy Sierra . Aurora said "This is another case where they're saying, 'The cases we care about are

2989-582: The last survey done of the gender balance in open source had been completed in 2006. Research methodology and a new survey were produced in 2011. A repeat of the survey took place in 2013, with hopes to provide a standard resource for the industry. The 2011 survey invited participants of any gender and inquired about subjects regarding open source and free software, hardware, open mapping , and other related open source areas, as well as free culture such as Creative Commons , online activism, mashup , maker , hacker spaces and related communities. The Ada Initiative

3050-396: The neighborhood f ( U ) {\displaystyle f(U)} is not required to be an open neighborhood). Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection f : ( X , τ ) → ( Y , σ ) {\displaystyle f:(X,\tau )\to (Y,\sigma )}

3111-512: The ones white men are interested in. We’re less interested in protecting women on the web.'" This comment was received negatively by the EFF's Director of International Freedom of Expression, Jillian York . Another 2013 controversy that received commentary from Aurora was Donglegate in which a female attendee at a nearly all-male PyCon attendee faced backlash for reporting a conversation overheard between two men sitting near her. Aurora condemned

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3172-495: The organization, including changing its programs, or becoming volunteer-only. After much deliberation, the board decided to do an orderly shutdown of the Ada Initiative, in which the organization would open source all of our remaining knowledge and expertise in freely reusable and modifiable form. We don’t feel like non-profits need to exist forever. The Ada Initiative did a lot of great work, and we are happy about it. The previous hire of Crystal Huff, announced several months earlier,

3233-418: The projection X × Y → X {\displaystyle X\times Y\to X} is closed. This is essentially the tube lemma . To every point on the unit circle we can associate the angle of the positive x {\displaystyle x} -axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval [0,2π)

3294-525: The projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection p 1 : R 2 → R {\displaystyle p_{1}:\mathbb {R} ^{2}\to \mathbb {R} } on the first component; then the set A = { ( x , 1 / x ) : x ≠ 0 } {\displaystyle A=\{(x,1/x):x\neq 0\}}

3355-422: The same interview, she approved of the culture of the website Tumblr and stated that Linus Torvalds' daughter Patricia was a positive role-model. Open mapping In mathematics , more specifically in topology , an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X → Y {\displaystyle f:X\to Y}

3416-477: The threats sent to the woman and stated that Anonymous , by using large numbers of computers, was "distorting social pressure". When asked if firing one of the males was an appropriate response, she said "I don't have enough information to know that." Two years later, Aurora praised the gender ratio at PyCon and called Guido van Rossum and the Python community "the biggest success story for women in open source." In

3477-498: The word "closed". A map f : X → Y {\displaystyle f:X\to Y} is called an open map or a strongly open map if it satisfies any of the following equivalent conditions: If B {\displaystyle {\mathcal {B}}} is a basis for X {\displaystyle X} then the following can be appended to this list: A map f : X → Y {\displaystyle f:X\to Y}

3538-623: Was Linux Australia , who provided support alongside Puppet Labs , DreamHost , The Mail Archive and Google . Aurora and Gardiner were the only staff members, serving full-time roles in the organization. The Ada Initiative was governed by a seven-person board of directors , who oversaw its management. The board included co-founder Mary Gardiner , Sue Gardner , Amelia Greenhall , Rachel Chalmers, Alicia Gibb, Andrea Horbinski and Marina Zhurakhinskaya. An advisory board of about 30 members provided input about ideas and projects. In collaboration with members of LinuxChix , Geek Feminism and other groups,

3599-418: Was not mentioned other than to note "that hire didn't work out." All services provided by the Ada Initiative were pro bono , and the organization was supported by member donations. In the summer of 2011, the Ada Initiative launched a campaign to raise start-up funds with a goal of contributions from 100 funders. The campaign wrapped up six days before its planned deadline. The organization's first major sponsor

3660-581: Was raised in New Mexico , and was home-schooled. She became involved in computer programming when she attended DEF CON in 1995. She studied computer science and mathematics at the New Mexico Institute of Mining and Technology . She first became involved with file systems when she worked with ZFS in 2002 at Sun Microsystems . She later moved to IBM where she worked in the group of Theodore Ts'o , where they considered extensions to

3721-519: Was the organizer of AdaCamp , an unconference "dedicated to increasing women’s participation in open technology and culture." Seven AdaCamps were held between 2012 and 2015. In February 2013, the organizers of the Security B-Sides San Francisco conference canceled speaker Violet Blue 's talk, sex +/- drugs: known vulns and exploits, due to concerns raised by the Ada Initiative that it contained rape triggers , as well as

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