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Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space, homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric d: X × X → [ 0, ∞ ) such that the topology induced by d is T. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited; this is true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. One of the first recognized metrization theorems was Urysohn's metrization theorem; this states. So, for example, every second-countable manifold is metrizable.. The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.

The Nagata–Smirnov metrization theorem, described below, provides a more specific theorem where the converse does hold. Several other metrization theorems follow as simple corollaries to Urysohn's theorem. For example, a compact Hausdorff space is only if it is second-countable. Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular and second-countable; the Nagata–Smirnov metrization theorem extends this to the non-separable case. It states that a topological space is metrizable if and only if it is regular and has a σ-locally finite base. A σ-locally finite base is a base, a union of countably many locally finite collections of open sets. For a related theorem see the Bing metrization theorem. Separable metrizable spaces can be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube N, i.e. the countably infinite product of the unit interval with itself, endowed with the product topology. A space is said to be locally metrizable.

Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and paracompact. In particular, a manifold is only if it is paracompact; the group of unitary operators U on a separable Hilbert space H endowed with the strong operator topology is metrizable. Non-normal spaces cannot be metrizable; the real line with the lower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology; this space is Hausdorff and first countable. The long line is not metrizable. Uniformizability, the property of a topological space of being homeomorphic to a uniform space, or equivalently the topology being defined by a family of pseudometrics Moore space Apollonian metric Nagata–Smirnov metrization theorem Bing metrization theorem This article incorporates material from Metrizable on PlanetMath, licensed under the Creative Commons Attribution/Share-Alike License

2001 Japan Series

The 2001 Japan Series was the 52nd edition of Nippon Professional Baseball's postseason championship series. It matched the Central League champion Yakult Swallows against the Pacific League champion Osaka Kintetsu Buffaloes; the Swallows defeated the Buffaloes in five games to claim their fifth Japan Series championship. Kintetsu had one of the most powerful offenses seen in the league. Foreign import Tuffy Rhodes teamed up with Norihiro Nakamura to become one of the most feared hitting tandems in Nippon Professional Baseball history. Rhodes hit 55 home runs to tie the NPB record for most home runs hit in a season, while Nakamura hit 46 home runs of his own. Most of the core team from the 1997 Japan Series championship still remained in Kazuhisa Ishii and all-world catcher Atsuya Furuta. Shinya Miyamoto anchored the middle of the infield. Game 1 Game 1 would become one of the best pitching performances turned in by any one starter in quite some time. In fact, it became historic. Yakult ace Kazuhisa Ishii kept the powerful Buffaloes lineup at bay, striking out 12 batters and walking 4.

After walking the first batter of the game, Ishii retired the next 8 batters. He would no-hit the Buffaloes through 6 and 1/3 innings and shut them out, throwing a complete game. 2001 World Series

Tina Hobley

Tina Ellen Hobley is an English actress and radio presenter, best known for her long-running role as Chrissie Williams in the BBC One medical drama series Holby City. Hobley left Holby City in November 2013 after 12 years. Hobley was born in Hampstead, she was shy as a child, was sent to speech and drama classes in an attempt to counter her introversion. She attended Bishop Douglass School in East Finchley. Hobley transferred to the Webber Douglas Academy of Dramatic Art. After graduating from the Webber Douglas Academy of Dramatic Art, where she trained from 1990 to 1993, Hobley had a number of roles in a variety of television dramas, including Coronation Street as Samantha Failsworth, Harbour Lights as WPC Melanie Rush and The Bill as Sue Booker. Hobley is best known for her role as ward sister Chrissie Williams in BBC medical drama Holby City. On 3 October 2013, Smooth Radio announced that Hobley would become a presenter on the network, fronting a programme on Sunday mornings. Hobly presented The Smooth Drive Home on Smooth London from January 2017 until July 2019.

From January to July 2015 Hobley co-starred with Jamie Lomas, Rik Makarem, Michael McKell and Gray O'Brien in a touring production of Peter James's "Dead Simple". Hobley's first marriage was to graphic designer Steve Wallington in 1998, their daughter Isabella,'Bella', was born in April 1999. On 22 April 2006, Hobley announced her engagement to Oliver Wheeler, they married that year. Hobley gave birth to their daughter Olivia on 18 April 2008, on 1 March 2010 to son, Orson. Hobley is a supporter of the Starlight Foundation, she is an Ambassador for Barnardo's, supports the Terrence Higgins Trust and White Hat Rally. In February 2016 she withdrew from competing in the reality show The Jump after dislocating her shoulder, breaking her arm and rupturing her anterior cruciate ligament. Seven months after sustaining these injuries she still had not recovered; the damage to her body had limited her movement to the point where she has been unable to perform basic daily tasks by herself. Guest appearances2003 TV Moments - Audience member This Morning Hell's Kitchen - Episode No. 1.10 The Paul O'Grady Show - Episode No. 1.22 The Xtra Factor GMTV Comic Relief Does Fame Academy Test the Nation: The Big Entertainment Test School's Out - Episode No. 2.3 The One Show - Guest All Star Mr & Mrs - Contestant with husband, Oli Lorraine - Guest Daybreak - Guest Big Star's Little Star - Contestant with daughter, Olivia The Paul O'Grady Show Loose Women Official website Tina Hobley on Smooth Radio Tina Hobley on IMDb

Frame analysis

Frame analysis is a multi-disciplinary social science research method used to analyze how people understand situations and activities. Frame analysis looks at images, stereotypes metaphors, actors and more, it examines how and why they are chosen. The concept is attributed to the work of Erving Goffman and his 1974 book Frame analysis: An essay on the organization of experience and has been developed in social movement theory, policy studies and elsewhere. Framing theory and frame analysis is a broad theoretical approach, used in communication studies, news and social movements among other applications. "Framing is the process by which a communication source, such as a news organization and constructs a political issue or public controversy". It is related to the concept of agenda-setting. Framing influences how people process information; this can set an agenda. However, frame analysis goes beyond agenda-setting by examining the issues rather than the topics. Frame analysis is done in regard to news media.

However, framing is inevitable. It can speed up the process of interpretation as well as presenting the news. People just may not realize; when people are aware that they are using framing, there are several techniques. These may include: metaphor, tradition, jargon, artifact, contrast or spin. Frame analysis had been proposed as a type of rhetorical analysis for political actors in the 1980s. Political communication researcher Jim A. Kuypers first published his work advancing framing analysis as a rhetorical perspective in 1997, his approach begins inductively by looking for themes that persist across time in a text, determining how those themes are framed. Kuypers' work begins with the assumption that frames are powerful rhetorical entities that "induce us to filter our perceptions of the world in particular ways making some aspects of our multi-dimensional reality more noticeable than other aspects, they operate by making some information more salient than other information...." In "Framing Analysis From a Rhetorical Perspective" Kuypers details the differences between framing analysis as rhetorical criticism and as a social scientific endeavor, in particular arguing that framing criticism offers insights unavailable to social scientists.

In his 2009 work, Rhetorical Criticism: Perspectives in Action Kuypers offers a detailed template for doing framing analysis from a rhetorical perspective. According to Kuypers, "Framing is a process whereby communicators, consciously or unconsciously, act to construct a point of view that encourages the facts of a given situation to be interpreted by others in a particular manner. Frames operate in four key ways: they define problems, diagnose causes, make moral judgments, suggest remedies. Frames are found within a narrative account of an issue or event, are the central organizing idea." Kuypers' work is based on the premise that framing is a rhetorical process and as such it is best examined from a rhetorical point of view. In his book, Goffman said. There are distinctions within primary frameworks. There are social frameworks. Natural frameworks don't apply social forces to situations, they just exist naturally. However, social frameworks do apply social forces to situations; the two are connected.

Framing has been utilized to explain the process of social movements. Movements are carriers of ideologies. In addition, they are part of the process of constructing meaning for opposers. Mass movements are said to be successful when the frames projected align with the frames of participants to produce resonance between the two parties; this is a process known as frame alignment. Snow and Benford say that frame alignment is an important element in social mobilization or movement, they argue that when individual frames become linked in congruency and complementariness, that "frame alignment" occurs, producing "frame resonance", key to the process of a group transitioning from one frame to another. The conditions that affect or constrain framing efforts are: "The robustness and thoroughness of the framing effort". Snow, Rochford and Benford identify three core framing tasks and the degree to which these tasks are attended to will determine participant mobilization; the three tasks are: diagnostic framing for the identification of a assignment of blame.

The relationship between the proposed frame and the larger belief system. Its range and interrelatedness – if the frame is linked to only one core belief or value that, in itself, is of limited range within the larger belief system, the frame has a high degree of being discounted. Relevance of the frame to the realities of the participants. Relevancy can be constrained by empirical credibility or testability, it relates to participant experience, has narrative fidelity, that is, it fits in with existing cultural myths and


WKJM is an Urban Adult Contemporary formatted broadcast radio station licensed to Petersburg, serving Petersburg, Colonial Heights, Chesterfield in Virginia. WKJM is operated by Radio One; the station's studios and offices are located just north of Richmond proper on Emerywood Parkway in unincorporated Henrico County, its transmitter is located in Petersburg. 99.3 dates back to October 1966, when it signed on as Petersburg based WSSV-FM, simulcasted the Top 40 format of sister station WSSV-AM. In 1970, then-owner Roger Bean, in order to concentrate on his more profitable cable TV operations, decided to sell the radio stations. WSSV AM/FM were purchased by Eure Communications, headed by William L. Eure. In 1973, responding to changes in FCC regulations regarding AM/FM simulcasts, the FM station began a separately programmed automated Beautiful Music format under the call letters WPLZ, with the slogan "Music To Please". In 1979, the station switched formats to Automated Top 40 as "99Z FM". In 1981, seeing that there was not an Urban station on the FM band in Central Virginia, the station flipped formats once again, became the first Urban-formatted FM in Central Virginia as "Magic 99FM".

In 1986, Eure Communications sold WPLZ-FM and WSSV-AM to Paco-John Broadcasting, headed by Philadelphia attorney Glenn Mahone, for $6.5 million. In November 1987, WCDX, through two previous unsuccessful formats, changed to an Urban format. WCDX's stronger signal in Richmond caused WPLZ ratings to drop. Paco-John, to compensate, purchased the 99.3 frequency in Fredericksburg and became a simulcast on both 99.3 frequencies, which brought WPLZ's programming in the northern areas of the Richmond Metro, including Hanover County and parts of Henrico. This simulcast, for a time, gave not only Fredericksburg an Urban station, but Charlottesville one as well. By the early 1990s, despite the simulcasting, WPLZ was unable to regain its lead over WCDX. Paco-John attempted to expand its holdings by attempting to purchase WMYK in Norfolk later WGH-AM/FM. In both cases, the company was unable to secure financing for the purchases and the sales were cancelled. Paco-John went into bankruptcy and the stations were put under control of a trustee named Charles Giddens.

Both of the 99.3 stations were sold, in May 1992, WCDX's owner Sinclair Telecable ended up purchasing WPLZ and its AM sister station, which by this time had become gospel WGCV, moving the studio's and offices out of Petersburg to their current location on Emerywood Parkway in Richmond. At first, WPLZ's format was adjusted to Urban AC, but in December 1998, it switched to Urban Oldies as "Magic 99 FM". By the end of the 1990s, Radio One had purchased not only WPLZ and WGCV, but WCDX, WJRV, WARV, WKJS. In March 2001, Radio One moved WPLZ and its Urban Oldies format to 105.7, while simulcasting with 99.3. A month a gospel format debuted on 99.3 as WPZE. Getting advance warning of a format flip to Urban getting ready to happen on Clear Channel's WRCL, on June 9, 2001, Radio One flipped 99.3 to urban as WRHH, "Hot 99.3", just one day ahead of the 106.5 format flip. On December 3, 2003, 99.3 switched back to a gospel format as WPZZ, "Praise 99.3". On November 18, 2004, the "Praise" format moved to 104.7 FM, with 99.3 joining in a simulcast with 105.7 FM and adopted an Urban AC format as "Kiss FM."

99.3 and 105.7 Kiss FM Online Query the FCC's FM station database for WKJM Radio-Locator information on WKJM Query Nielsen Audio's FM station database for WKJM

1952 United States presidential election in North Dakota

The 1952 United States presidential election in North Dakota took place on November 4, 1952, as part of the 1952 United States presidential election. Voters chose four representatives, or electors, to the Electoral College, who voted for president and vice president. North Dakota was won by Columbia University President Dwight D. Eisenhower, running with California Senator Richard Nixon, with 70.97%t of the popular vote, against Adlai Stevenson, running with Alabama Senator John Sparkman, with 28.39% of the popular vote. As of the 2016 presidential election, this is the last election in which Rolette County voted for a Republican Presidential candidate; this would be the final time until 1988 when national bellwether Sargent County in the state's southeast would support a losing candidate,With 70.97% of the popular vote, North Dakota was Eisenhower's second strongest state after Vermont