Talk:Retraction (topology) and The Crofoot: Difference between pages

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'''History of The Crofoot'''
{{maths rating
| field = topology
| importance = low
| class = start
| historical =


Pontiac is the first inland city of Michigan. The Pontiac Company platted it in 1818 where the Saginaw Indian Trail (Woodward Avenue) crossed the Schiawassee Indian Trail (Orchard Lake Road). The Clinton River created double peninsulae where it snaked through what would become downtown Pontiac. The Pontiac Commercial Historic District is located just north of where the Saginaw Trail crossed the Clinton River. Historically, the business district of Pontiac was centered at the intersection of Pike and Saginaw.
}}


On April 30, of 1840, the entire commercial district on both sides of Saginaw from Pike to Lawrence was leveled in a fire. In response to this event, all new buildings were subsequently built with masonry construction. Between 1840 and the end of the Civil War, much of the buildings along Saginaw had been rebuilt.
Hmmmm...
In the 1970’s the entire area on the South West quadrant of Pike and Saginaw, and much of the historic property to the south of the Clinton River (Water Street), were demolished for urban renewal (Some members of the Oakland County Historic Society still sadly refer to that area as “Hiroshima Flats”.).
[[Image:Crofoot block pike and saginaw 1928.jpg|thumb|Crofoot Block 1928 with Third Floor Copula|200px|right|Crofoot Block 1928]]
Image showing 10'X10' third floor north facing window


The Crofoot Project has escaped both of these tragedies of Pontiac History. The building on the SE corner of Pike and Saginaw was not burned down. And it was spared demolition in the 1970’s.
"a continuous map r is a retract if ... In this case, A is called a retract of X, and r is called a retraction"
In 1882, it was rebuilt, re-using the existing foundations, floors and some walls.
Michael E Crofoot, who named the subject property the “Crofoot Block”, was a vigorous and active man whose life epitomized the development of Michigan and Pontiac after its 1818 founding, in the Civil War era, prior to the rapid grown from the expansion of the automobile industry.


Michael E. Crofoot was a prominent businessman, attorney, Judge of Oakland County Probate Court from 1849 to 1856, and a man involved in Oakland County, Michigan and national affairs.
make up your mind, dudes
He was a delegate to the 1856 Democratic National Convention. After the Civil War, he was selected in 1865, to represent Oakland County in raising subscriptions for the Soldiers and Sailors Monument in Downtown Detroit. He lived three blocks up the hill on Williams Street, where his home was one of 9 Pontiac landmarks featured on an 1867 “Birds Eye View of Pontiac” by Albert Ruger. He was on the State Building Commission for the State Asylum in 1874, which was completed in Pontiac in 1878. He represented a defendant in 1882 before the US Supreme Court. He rebuilt the 1830 era building at Pontiac’s first corner in 1882 and named it the “Crofoot Block”. He practiced law on the 3rd floor, overlooking from his 10 foot by 10 foot north-facing window the rapid growth of Pontiac’s Downtown Commercial District.
Pontiac’s Crofoot School was named and this prominent family, and is still in use.
The Crofoot Project, consisting of three adjacent buildings: the Crofoot Building, the Vernor's Building, and the New Crofoot Block, is a unique contributor to the local Pontiac Commercial Historic District.


The Crofoot Building, located at the crossroads of Pontiac's original 1818 plat, has anchored the Southeast quadrant of Pontiac since 1882. The building's central tower once displayed, in raised masonry letters, the name of its builders, the locally prominent Crofoot family. The Crofoot Building's three 20 foot wide bays and two stories of street-level store and second-floor office uses originally formed and still retain the urban pattern of 19th century Pontiac. Facade remodelings and occupant turnovers have little changed this pattern of uses. The Crofoot Building's street level has housed barbers, meat markets, an American Express office, saloons, lunch rooms, shoe and millinery shops; while the second floor has housed photographers, land developers, tailors, insurance agencies, and attorneys. The third floor (no longer extant) once housed Judge Michael E. Crofoot's legal offices.
:Fixed. Apart from the repetition, the map is called a ''retraction'', while the subspace is a ''retract''. There was some confusion about it. [[User:Alvisetrevi|Alvisetrevi]] 10:25, 27 September 2007 (UTC)


The Crofoot Building is arguably the largest physical survivor of Pontiac's original plat and of its brick nineteenth century architecture. The building has outlasted both of the 1920’s expansive remodeling boom and the 1970's demolitions in the name of urban renewal.
== strong deformation retract? ==


The Vernor's Building illustrates another thread in downtown Pontiac's urban formation: the single 20-feet wide Victorian facade completely remodeled in the 1920s boom. The Vernor's Building originally shared a similar Victorian facade with the Crofoot Building: open storefronts at street level, a brick second floor pierced by windows, and a third-floor, full-height mansard roof (Photo 3). In 1926, the building was remodeled to house a Vernor's Ginger Ale Soda Fountain; and its distinct brick second-floor was obliterated in a quickening of urban intensity, to form a two-level open storefront (Photo 2 ). Its mansard roof was extant until at least 1935; it was lost sometime after, perhaps when the Crofoot Building lost its third-floor mansard roof and tower. The Vernor's Soda Fountain remained at this address for at least twenty years.
I've seen the phrase "strong deformation retract". How's that related to "deformation retract"? thanks! --[[User:345Kai|345Kai]] 16:10, 5 March 2007 (UTC)


The New Crofoot Block dates not from Victorian downtown Pontiac, but from the expansive automobile-manufacturing Pontiac of the early twentieth century. The New Crofoot Block occupies the rear of Lot 59, the original rear yard of the Crofoot Building, which was earlier occupied by a service barn. By c. 1912, this rear yard had become too valuable for such a use, and a new two-story and basement structure was built, filling the lot. The building's street level had four 17-feet wide storefronts facing Pike Street, and the second floor had office space accessible through the original Crofoot Building. The Pontiac City Directory referred to these second-floor offices as "The New Crofoot Block".
== Definition ==


The architecture of the new building was a business-like and contemporary, rather than an attempt to emulate its Victorian namesake. The new facade emphasized its large window openings on both floors, surrounding them with a dark cider-colored brick frame, trimmed with limestone, and iced on the top with a pressed-metal entablature.
According to the history, people seem to keep changing the third line of the definition from d(a,t)=a (correct) to d(a,1)=a (true but incorrect for the definition). Most sources (eg. Hatcher) seem to favour the first definition, but more importantly, the second one is inconsistent with the claim at the end that "contractible spaces exist which do not deformation retract to a point", and also makes defining t redundant. Please stop changing it back - it stumped me for a good 15 minutes and I'm sure it'll stump other assignment-doers too =P [[User:Pirsq|Pirsq]] 11:58, 22 June 2007 (UTC)
: Well, I don't think that choosing a definition according to whether or not it contrasts with other claims in a Wikipedia page is a good idea... suppose we choose to define a ''deformation retract'' using d(a,1)=a and a ''strong deformation retract'' using d(a,t)=a. In this case we could modify the claim to "contractible spaces exist which do not strongly deformation retract to a point" (or something like that) and we're all set! The definition with d(a,1)=a has the advantage, in my opinion, of being equivalent to the following concise and elegant definition:
::Let <math>\iota: A \hookrightarrow X</math> be the inclusion. A '''deformation retract'' <math>r</math> is a retract such that <math>\iota \circ r</math> is homotopic to the identity of ''X''.
:This has a one-line proof! Another advantage is that the homotopy equivalence of ''A'' and ''X'' is immediate. If no one disagrees within a couple of days I will edit the entry.[[User:Alvisetrevi|Alvisetrevi]] 20:34, 26 September 2007 (UTC)


These three adjacent buildings, despite their different origins, were joined together sometime after World War II. Their heights were equalized, their second-floor windows were ruthlessly reset, and their combined street facades were wrapped in a skin of porcelain-metal panels (Photos 6,7), exemplifying another stage in the development of downtown Pontiac.
Then people seem to be using the following definitions: [http://mathworld.wolfram.com/StrongDeformationRetract.html strong deformation retract], [http://mathworld.wolfram.com/DeformationRetract.html deformation retract]. In most sources I have seen these two definitions are used, but haven't checked enough of them to argue about sources in general. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/83.9.83.147|83.9.83.147]] ([[User talk:83.9.83.147|talk]]) 18:52, 25 September 2007 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->


The Crofoot Project is a unique contributor to the local Pontiac Commercial Historic District because: 1) it is an assemblage of three distinct adjoining buildings that exemplify three threads in the development of the District; 2) the Crofoot Building is unique in its crossroads location, its size, footprint, and date in the District; 3) the three buildings have survived with their footprints intact; 4) the three buildings have retained their historic and urban layering of street-level retail and second-floor office uses; and 5) all three buildings have retained large portions of their historic exterior integrity.
Sorry, I just caught up with my watchlist and saw this. After checking four or five of the "standard" textbooks, it appears that there is disagreement about definitions. The ideal article would not only mention the prevailing definitions, citing the textbooks that use them, but would also address the issue of which definitions are needed for which results. (Certain statements are only true using the "stronger" form of deformation, for example.) It might make for a little more bulk than we would want, but we can't just sweep this issue under the rug. Now that Hatcher is fast becoming the go-to textbook, I want to be especially careful that we don't ignore the definitions there, which differ from the ones here. I am swamped for the moment, but I'll try to come back to this if I get time. [[User:VectorPosse|VectorPosse]] 16:40, 1 October 2007 (UTC)
<!-- Image with unknown copyright status removed: [[Image:Crofoot block pike and saginaw today.jpg|thumb|Crofoot Block today with Third Floor gone|200px|right|Crofoot Block today]] -->


'''Crofoot Today'''
:Yes, what you say definitely makes sense. Of course that needs time, and time is money!. What follows is just a "bibliographical" note. From my memory, Hatcher and May (A concise course in AT) use the stronger version, while Bredon, Bott&Tu and Rotman (An intro to AT) use the weaker one. I just checked Massey: he goes for the stronger one, but mentions the existence of different uses. [[User:Alvisetrevi|Alvisetrevi]] 08:38, 2 October 2007 (UTC)
As recently as 2005, the City of Pontiac had condemned this property, and scheduled it for demolition. In 2006 a local developer began the renovation of this important Pontiac landmark. On Sept 7, 2007 the doors to The Crofoot reopened as a new home for music & art in Metro Detroit. The building features a state of the art sound system, and much of the original historical intergrety remains. In the first year of shows the Crofoot has played host to more than 300 live events, including shows from [[Jimmy Eat World]], [[The National (band)]], [[Girl Talk (musician)]],[[The New Pornographers]], [[Craig Owens (musician)]], [[Stars (band)]], [[Band of Horses]], [[Vampire Weekend]] and [[Allan James and the Cold Wave]]...


{{coord missing|United States}}
== Example? ==
"However, there exist contractible spaces which do not strongly deformation retract to a point."
An example of this here would be fantastic. [[User:Msgj|msgj]] 09:18, 11 October 2007 (UTC)


[[Category:Buildings and structures in Michigan|Crofoot]]

[[Category:Pontiac, Michigan]]
I've been puzzeling over this for some time and an example is the line with two origins. It can be retracted to the top origin, but during the courser of this deformation the bottom origin needs to leave the origin to get to the top origin, on the way it has to take the top origin with it, thus it cannot be strongly deformed to the top origin. -- This is very informal outline of a proof , I hope its good enough to help. -cwd1 <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Cwd1|Cwd1]] ([[User talk:Cwd1|talk]] • [[Special:Contributions/Cwd1|contribs]]) 17:35, 11 October 2007 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->


I couldn't make sense of this example... sorry, maybe I am not used to working with this notion. An example I could came up with is the following:
take the subset of the square IxI defined as: I x ( {0} U {1/n : n is a natural number} ) -- this is the union of the base horizontal segment and countably many vertical segments leaving from the base, one for each 1/n, plus the vertical segment corresponding to the left side of the square. Try to draw it, it is much easier to do so than to describe it! It is some sort of "infinite comb". There is a retraction to the top left corner {0,1} of the square: a point {1/n,y} in a tooth of the comb is firs sent down the tooth, then left until the side of the square and finally up on the side to {0,y}, to the same height it was before. This is clearly a deformation retract, but it can't be strong: the point {0,1} on the left side has to go down to {0,0} and then up again. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Alvisetrevi|Alvisetrevi]] ([[User talk:Alvisetrevi|talk]] • [[Special:Contributions/Alvisetrevi|contribs]]) 08:48, 24 April 2008 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== Retraction v/s Idempotency ==

How is a retraction different from an [[Idempotence|idempotent]] map on <math>X</math>? If <math>r: X \rightarrow A</math> is a retraction then <math> r(r(x)) = r(x) </math> because <math>r(x) \in A</math>. OTOH, if <math>r</math> is idempotent then for all <math>a</math> in the codomain of <math>r</math> there's some <math>x</math> such that <math>r(x)=a</math> therefore <math>r(a)=r(r(x))=r(x)=a</math>. The two definitions seem therefore equivalent, yet the respective articles don't even mention each other. <span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/192.114.107.4|192.114.107.4]] ([[User talk:192.114.107.4|talk]]) 07:54, 16 September 2008 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->


Well, what you say would be true only if your idempotent map was 1) surjective 2) continuous. Indeed, you're stating "for all <math>a</math> in the codomain of <math>r</math> there's some <math>x</math> such that <math>r(x)=a</math>". But what if not all the elements of <math>A</math> lie in the codomain of <math>r</math>?. A practical example: the absolute value from complex numbers to real numbers. It is idempotent, but definitely not a retraction (e.g. <math>r(-5)=5 \neq -5 </math>).
[[User:Alvisetrevi|Alvisetrevi]] ([[User talk:Alvisetrevi|talk]]) 21:37, 10 October 2008 (UTC)

Revision as of 00:33, 11 October 2008


History of The Crofoot

Pontiac is the first inland city of Michigan. The Pontiac Company platted it in 1818 where the Saginaw Indian Trail (Woodward Avenue) crossed the Schiawassee Indian Trail (Orchard Lake Road). The Clinton River created double peninsulae where it snaked through what would become downtown Pontiac. The Pontiac Commercial Historic District is located just north of where the Saginaw Trail crossed the Clinton River. Historically, the business district of Pontiac was centered at the intersection of Pike and Saginaw.

On April 30, of 1840, the entire commercial district on both sides of Saginaw from Pike to Lawrence was leveled in a fire. In response to this event, all new buildings were subsequently built with masonry construction. Between 1840 and the end of the Civil War, much of the buildings along Saginaw had been rebuilt. In the 1970’s the entire area on the South West quadrant of Pike and Saginaw, and much of the historic property to the south of the Clinton River (Water Street), were demolished for urban renewal (Some members of the Oakland County Historic Society still sadly refer to that area as “Hiroshima Flats”.).

Crofoot Block 1928

Image showing 10'X10' third floor north facing window

The Crofoot Project has escaped both of these tragedies of Pontiac History. The building on the SE corner of Pike and Saginaw was not burned down. And it was spared demolition in the 1970’s. In 1882, it was rebuilt, re-using the existing foundations, floors and some walls. Michael E Crofoot, who named the subject property the “Crofoot Block”, was a vigorous and active man whose life epitomized the development of Michigan and Pontiac after its 1818 founding, in the Civil War era, prior to the rapid grown from the expansion of the automobile industry.

Michael E. Crofoot was a prominent businessman, attorney, Judge of Oakland County Probate Court from 1849 to 1856, and a man involved in Oakland County, Michigan and national affairs. He was a delegate to the 1856 Democratic National Convention. After the Civil War, he was selected in 1865, to represent Oakland County in raising subscriptions for the Soldiers and Sailors Monument in Downtown Detroit. He lived three blocks up the hill on Williams Street, where his home was one of 9 Pontiac landmarks featured on an 1867 “Birds Eye View of Pontiac” by Albert Ruger. He was on the State Building Commission for the State Asylum in 1874, which was completed in Pontiac in 1878. He represented a defendant in 1882 before the US Supreme Court. He rebuilt the 1830 era building at Pontiac’s first corner in 1882 and named it the “Crofoot Block”. He practiced law on the 3rd floor, overlooking from his 10 foot by 10 foot north-facing window the rapid growth of Pontiac’s Downtown Commercial District. Pontiac’s Crofoot School was named and this prominent family, and is still in use. The Crofoot Project, consisting of three adjacent buildings: the Crofoot Building, the Vernor's Building, and the New Crofoot Block, is a unique contributor to the local Pontiac Commercial Historic District.

The Crofoot Building, located at the crossroads of Pontiac's original 1818 plat, has anchored the Southeast quadrant of Pontiac since 1882. The building's central tower once displayed, in raised masonry letters, the name of its builders, the locally prominent Crofoot family. The Crofoot Building's three 20 foot wide bays and two stories of street-level store and second-floor office uses originally formed and still retain the urban pattern of 19th century Pontiac. Facade remodelings and occupant turnovers have little changed this pattern of uses. The Crofoot Building's street level has housed barbers, meat markets, an American Express office, saloons, lunch rooms, shoe and millinery shops; while the second floor has housed photographers, land developers, tailors, insurance agencies, and attorneys. The third floor (no longer extant) once housed Judge Michael E. Crofoot's legal offices.

The Crofoot Building is arguably the largest physical survivor of Pontiac's original plat and of its brick nineteenth century architecture. The building has outlasted both of the 1920’s expansive remodeling boom and the 1970's demolitions in the name of urban renewal.

The Vernor's Building illustrates another thread in downtown Pontiac's urban formation: the single 20-feet wide Victorian facade completely remodeled in the 1920s boom. The Vernor's Building originally shared a similar Victorian facade with the Crofoot Building: open storefronts at street level, a brick second floor pierced by windows, and a third-floor, full-height mansard roof (Photo 3). In 1926, the building was remodeled to house a Vernor's Ginger Ale Soda Fountain; and its distinct brick second-floor was obliterated in a quickening of urban intensity, to form a two-level open storefront (Photo 2 ). Its mansard roof was extant until at least 1935; it was lost sometime after, perhaps when the Crofoot Building lost its third-floor mansard roof and tower. The Vernor's Soda Fountain remained at this address for at least twenty years.

The New Crofoot Block dates not from Victorian downtown Pontiac, but from the expansive automobile-manufacturing Pontiac of the early twentieth century. The New Crofoot Block occupies the rear of Lot 59, the original rear yard of the Crofoot Building, which was earlier occupied by a service barn. By c. 1912, this rear yard had become too valuable for such a use, and a new two-story and basement structure was built, filling the lot. The building's street level had four 17-feet wide storefronts facing Pike Street, and the second floor had office space accessible through the original Crofoot Building. The Pontiac City Directory referred to these second-floor offices as "The New Crofoot Block".

The architecture of the new building was a business-like and contemporary, rather than an attempt to emulate its Victorian namesake. The new facade emphasized its large window openings on both floors, surrounding them with a dark cider-colored brick frame, trimmed with limestone, and iced on the top with a pressed-metal entablature.

These three adjacent buildings, despite their different origins, were joined together sometime after World War II. Their heights were equalized, their second-floor windows were ruthlessly reset, and their combined street facades were wrapped in a skin of porcelain-metal panels (Photos 6,7), exemplifying another stage in the development of downtown Pontiac.

The Crofoot Project is a unique contributor to the local Pontiac Commercial Historic District because: 1) it is an assemblage of three distinct adjoining buildings that exemplify three threads in the development of the District; 2) the Crofoot Building is unique in its crossroads location, its size, footprint, and date in the District; 3) the three buildings have survived with their footprints intact; 4) the three buildings have retained their historic and urban layering of street-level retail and second-floor office uses; and 5) all three buildings have retained large portions of their historic exterior integrity.

Crofoot Today As recently as 2005, the City of Pontiac had condemned this property, and scheduled it for demolition. In 2006 a local developer began the renovation of this important Pontiac landmark. On Sept 7, 2007 the doors to The Crofoot reopened as a new home for music & art in Metro Detroit. The building features a state of the art sound system, and much of the original historical intergrety remains. In the first year of shows the Crofoot has played host to more than 300 live events, including shows from Jimmy Eat World, The National (band), Girl Talk (musician),The New Pornographers, Craig Owens (musician), Stars (band), Band of Horses, Vampire Weekend and Allan James and the Cold Wave...