Square-lattice Ising model

In mathematics, in the field of topology, a perfect set is a closed set with no isolated points, and a perfect space is any topological space with no isolated points. In such spaces, every point can be approximated arbitrarily well by other points – given any point and any topological neighborhood of the point, there is another point within the neighborhood.

The term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a G_δ space. Context is required to determine which meaning is intended.

In this article, a space which is not perfect will be referred to as imperfect.

Examples and nonexamples

The real line $\mathbb{ℝ}$ is a connected perfect space, while the Cantor space 2^ω and Baire space ω^ω are perfect, totally disconnected zero dimensional spaces.

Any nonempty set admits an imperfect topology: the discrete topology. Any set with more than one point admits a perfect topology: the indiscrete topology.

Imperfection of a space

Define the imperfection of a topological space to be the number of isolated points. This is a cardinal invariant – i.e., a mapping which assigns to each topological space a cardinal number such that homeomorphic spaces get assigned the same number.

A space is perfect if and only if it has imperfection zero.

Closure properties

Every nonempty perfect space has subsets which are imperfect in the subspace topology, namely the singleton sets. However, any open subspace of a perfect space is perfect.

Perfection is a local property of a topological space: a space is perfect if and only if every point in the space admits a basis of neighborhoods each of which is perfect in the subspace topology.

Let $\{X_i{\}}_{i\in I}$ be a family of topological spaces. As for any local property, the disjoint union $\coprod _i{X}_i$ is perfect if and only if every $X_i$ is perfect.

The Cartesian product of a family $\{X_i{\}}_{i\in I}$ is perfect in the product topology if and only if at least one of the following holds:

(i) At least one $X_i$ is perfect.

(ii) $I=\mathrm{\varnothing }$.

(iii) The set of indices $i\in I$ such that $X_i$ has at least two points is infinite.

A continuous image, and even a quotient, of a perfect space need not be perfect. For example, let X = R − {0}, let Y = {1, 2} given the discrete topology and let f be a function defined such that f(x) = 2 if x > 0 and f(x) = 1 if x < 0. However, every image of a perfect space under an injective continuous map is perfect.

Connection with other topological properties

It is natural to compare the concept of a perfect space – in which no singleton set is open – to that of a T₁ space – in which every singleton set is closed.

A T₁ space is perfect if and only if every point of the space is an $\omega$-accumulation point. In particular a nonempty perfect T₁ space is infinite.

Any connected T₁ space with more than one point is perfect. (More interesting therefore are disconnected perfect spaces, especially totally disconnected perfect spaces like Cantor space and Baire space.)

On the other hand, the set $X=\{\circ ,\bullet \}$ endowed with the topology $\{\mathrm{\varnothing },\{\circ \},X\}$ is connected, T₀ (and even sober) but not perfect (this space is called Sierpinski space).

Suppose X is a homogeneous topological space, i.e., the group $\mathrm{Aut}(X)$ of self-homeomorphisms acts transitively on X. Then X is either perfect or discrete. This holds in particular for all topological groups.

A space which is of the first category is necessarily perfect (so, similar to compactifiying a space, we can 'make' a space to be of the second category by taking the disjoint union with a one-point space).

Perfect spaces in descriptive set theory

Classical results in descriptive set theory establish limits on the cardinality of non-empty, perfect spaces with additional completeness properties. These results show that:

• If X is a complete metric space with no isolated points, then the Cantor space 2^ω can be continuously embedded into X. Thus X has cardinality at least $2^{{\mathrm{\aleph }}_0}$. If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly $2^{{\mathrm{\aleph }}_0}$.
• If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X, and so X has cardinality at least $2^{{\mathrm{\aleph }}_0}$.

See also

• Dense-in-itself
• Finite intersection property
• Derived set (mathematics)
• Subspace topology

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010